Finding Modular Inverses

1. Formula
   Observation

   The definition line uses the integer symbol a in “integers a and b are multiplicative inverses modulo m if ab ≡ 1 (mod m).”

2. Formula
   Observation

   The theorem line uses the same symbol a in “The integer a has an inverse modulo m if and only if (a,m)=1.”

a

An integer whose multiplicative inverse modulo m is being discussed; in the examples it is instantiated as 5 or 6.

Integers.

1. Formula
   Observation

   The definition line uses b in “integers a and b are multiplicative inverses modulo m if ab ≡ 1 (mod m).”

b

An integer that serves as a multiplicative inverse of a modulo m when ab ≡ 1 (mod m).

Integers.

1. Formula
   Observation

   The definition line contains “modulo m” and “(mod m)”.

2. Formula
   Observation

   The theorem line contains “inverse modulo m” and “(a,m)=1”.

m

The modulus for the congruence relation and for the existence condition of the inverse.

Integers used as a modulus; the video does not explicitly state extra restrictions on m.

1. Formula
   Observation

   The theorem line shows “(a,m)=1”.

2. Audio
   Observation

   The speaker reads this as “the greatest common divisor of a and m is equal to 1.”

(a,m)

The greatest common divisor of a and m.

Defined for the integer pair a and m appearing in the theorem.

1. Formula
   Observation

   The definition uses “ab ≡ 1 (mod m)”.

2. Formula
   Observation

   The worked examples use chains such as “1 ≡ 8 ≡ 15 (mod 7)” and “1 ≡ 12 ≡ 23 ≡ 34 ≡ 45 (mod 11)”.

≡

Congruence modulo the stated modulus.

Used between integers with a specified modulus in parentheses.

1. Formula
   Observation

   The first example writes “$5^{-1}$ ≡ 3 (mod 7)”.

2. Formula
   Observation

   The second example writes “$5^{-1}$ ≡ 9 (mod 11)”.

3. Formula
   Observation

   The third example writes “$6^{-1}$ ≡ 3 (mod 17)”.

$x^{-1}$

Notation for the multiplicative inverse of x modulo the given modulus.

Used for integers x in the examples, namely 5 and 6.

1. Formula
   Observation

   The modulus 19 appears in the problem title and in congruence statements such as 18 ≡ -1 (mod 19) and 17 ≡ -2 (mod 19).

19

The modulus used throughout the main worked examples in this clip.

Positive integer modulus.

1. Formula
   Observation

   The handwritten symbol '≡' is used repeatedly to state congruences modulo 19.

≡

Congruence relation; a ≡ b (mod m) means a and b have the same remainder modulo m.

Binary relation on integers.

1. Formula
   Observation

   Notation such as $18^{-1}$, (-2)^{-1}, and $17^{-1}$ is written on the board.

$a^{-1}$

Multiplicative inverse of a modulo the stated modulus.

Defined when gcd(a,m)=1; here the relevant residues are invertible modulo 19.

1. Formula
   Observation

   The number 18 is the residue whose inverse modulo 19 is being found.

18

Residue class modulo 19; also identified as -1 modulo 19.

Integer representative of a residue class mod 19.

1. Formula
   Observation

   The second example asks for the inverse of 17 mod 19.

17

Residue class modulo 19; also identified as -2 modulo 19.

Integer representative of a residue class mod 19.

1. Formula
   Observation

   The board writes 18 ≡ -1 (mod 19).

-1

Negative residue equivalent to 18 modulo 19.

Residue class modulo 19.

1. Formula
   Observation

   On-screen text: “Defn: We say that integers a and b are multiplicative inverses modulo m if ab ≡ 1 (mod m).”

2. Audio
   Observation

   The speaker says, “So we're finding multiplicative inverses, and that's two numbers that multiply together to get you one mod m.”

The video defines a pair of integers a and b as multiplicative inverses modulo m exactly when their product is congruent to 1 modulo m. The speaker paraphrases this as “two numbers that multiply together to get you one mod m.”

1. a and b are integers.

2. The congruence is taken modulo m.

1. Formula
   Observation

   On-screen text: “Theorem: The integer a has an inverse modulo m if and only if (a,m)=1.”

2. Audio
   Observation

   The speaker says, “a has an inverse mod m if and only if the greatest common divisor of a and m is equal to 1.”

3. Audio
   Observation

   The speaker adds, “If the greatest common divisor between a and m is greater than 1, then there is no inverse mod m.”

The clip states a necessary-and-sufficient condition for the existence of a modular inverse: an integer a has an inverse modulo m precisely when gcd(a,m)=1. The speaker also states the contrapositive in words: if gcd(a,m)>1, then no inverse modulo m exists.

1. a is an integer.

2. m is the modulus.

3. (a,m) denotes gcd(a,m).

1. Definition of multiplicative inverses modulo m

1. Audio
   Observation

   In the first example the speaker says, “So we do as we've done in the past. Mod 7, 1 is congruent to 8. That's not a multiple of 5. So we add 7 again and we get 15.”

2. Audio
   Observation

   In the second example the speaker says, “Always start with 1 and add the mod on.”

3. Formula
   Observation

   The written chains are “1 ≡ 8 ≡ 15 (mod 7)”, “1 ≡ 12 ≡ 23 ≡ 34 ≡ 45 (mod 11)”, and “1 ≡ 18 (mod 17)”.

The worked examples use a trial method: start from 1 and repeatedly add the modulus m until reaching a number divisible by the target integer a. Once such a representative is found, factor it as a times another integer; that other integer is the inverse modulo m.

1. Used to find an inverse of a modulo m.

2. One looks among integers congruent to 1 modulo m for a multiple of a.

1. Definition of multiplicative inverses modulo m

1. Audio
   Observation

   The speaker repeatedly says 'find the inverse of 18 mod 19' and later 'find the inverse of 17 mod 19'.

2. Formula
   Observation

   The board writes conclusions such as $18^{-1}$ ≡ 18 and $17^{-1}$ ≡ 9 (mod 19).

For a residue a modulo m, its inverse $a^{-1}$ is the residue b such that ab ≡ 1 (mod m). The clip uses this notion directly when searching for numbers whose product with 18 or 17 is congruent to 1 modulo 19.

1. The inverse exists when a is relatively prime to m.

2. In this clip the modulus is $m=19$.

1. ≡
2. $a^{-1}$

1. Audio
   Observation

   The speaker says they can start with 1 ≡ 20, then keep adding 19 until reaching a multiple of 18.

2. Formula
   Observation

   The board writes 1 ≡ 20 ≡ 39 ≡ 58 ≡ 77 ≡ ... ≡ 324, then factors 324 as $18\cdot 18$.

One way to find an inverse modulo m is to list numbers congruent to 1 modulo m by repeatedly adding m, then stop when one of those numbers is divisible by a. In the example, the sequence 20, 39, 58, 77, ..., 324 consists of numbers congruent to 1 modulo 19, and $324 = 18\cdot 18$ gives the inverse.

1. Works whenever a has an inverse modulo m.

2. May be slow if many terms must be checked.

1. Modular multiplicative inverse
2. ≡

1. Audio
   Observation

   The speaker introduces 'a little bit of a shortcut' by noticing 18 is the same thing as negative 1 mod 19, and later 17 is the same thing as negative 2 mod 19.

2. Formula
   Observation

   The board writes 18 ≡ -1 (mod 19) and 17 ≡ -2 (mod 19).

Instead of working only with positive representatives, replace a by an equivalent smaller-magnitude residue modulo m. This often makes the inverse easier to recognize. Here 18 is replaced by -1 and 17 by -2, turning the problem into finding inverses of simpler residues.

1. The replacement must be congruent modulo the same m.

2. The resulting residue should still be invertible modulo m.

1. Modular multiplicative inverse
2. ≡

1. Audio
   Observation

   The speaker asks what the inverse of negative 1 is and answers that multiplying negative 1 by negative 1 gives positive 1.

2. Formula
   Observation

   The board shows 18 ≡ -1 (mod 19) and concludes $18^{-1}$ ≡ 18 (mod 19).

Because (-1)(-1)=1, the residue -1 modulo any m is always its own multiplicative inverse. Since 18 ≡ -1 (mod 19), it follows immediately that $18^{-1}$ ≡ 18 (mod 19).

1. Valid for any modulus m.

1. Modular multiplicative inverse
2. Use negative residues to simplify inverse problems

1. Audio
   Observation

   The speaker says they will add 19 to negative 10 because they want positive numbers.

2. Formula
   Observation

   The board converts (-2)^{-1} ≡ -10 (mod 19) into $17^{-1}$ ≡ 9 (mod 19).

After finding an inverse in negative form, add the modulus once or more to obtain the standard positive representative. Here -10 + $19 = 9$, so the inverse of 17 modulo 19 is reported as 9.

1. Used when a negative residue answer is acceptable algebraically but a positive representative is preferred.

1. ≡
2. Use negative residues to simplify inverse problems

1. Formula
   Observation

   Printed title: "Find the inverse of 18 mod 19."

2. Formula
   Observation

   Written lines include $18 \equiv -1 \pmod{19}$ and $18^{-1} \equiv 1 \pmod{19}$.

3. Formula
   Observation

   Written lines include $17 \equiv -2 \pmod{19}$, (-2)^{-1} $\equiv -10 \pmod{19}$, and $17^{-1} \equiv 9 \pmod{19}$.

This segment shows two completed modular-inverse computations modulo 19. For 18, the presenter rewrites 18 as -1 modulo 19 and concludes that its inverse is 1 modulo 19. For 17, the presenter rewrites 17 as -2 modulo 19, uses (-2)^{-1} $\equiv -10 \pmod{19}$, and then records $17^{-1} \equiv 9 \pmod{19}$. The audio adds that the same result for 17 could also be reached by continuing a longer list of congruences until reaching $17 \times 9 = 153$.

1. Modulus is 19.

2. The displayed method assumes the relevant inverses already exist modulo 19.

1. Meaning of a modular inverse in this lesson

1. Formula
   Observation

   The board repeatedly uses notation $a^{-1} \equiv b \pmod m$ in examples such as $18^{-1} \equiv 1 \pmod{19}$ and $17^{-1} \equiv 9 \pmod{19}$.

2. Audio
   Observation

   The speaker frames both examples as "find the inverse" problems modulo a given number.

In this video, "the inverse of a mod m" means a residue class x modulo m that serves as a multiplicative inverse of a modulo m. The worked examples use the notation $a^{-1} \equiv x \pmod m$. The clip does not separately state the defining equation in words, but the notation and examples make clear that the topic is multiplicative inversion in modular arithmetic.

1. Applies to integers a and positive modulus m.

2. Existence is not automatic; the later example shows failure when gcd(a,m) $\ne 1$.

1. Formula
   Observation

   The board writes (5,10)=5 NOT 1 and then No inverse.

2. Audio
   Observation

   The speaker says, "the greatest common divisor of 5 and 10 is 5, which is not 1. So the answer is that there is no inverse."

3. Audio
   Observation

   The speaker generalizes: "if there's a greatest common divisor greater than 1 between those two values, then there is no inverse."

The second example teaches a quick existence test before trying to compute an inverse. For 5 mod 10, the presenter computes (5,10)=5, notes that this is not 1, and concludes that 5 has no inverse modulo 10. The spoken rule is generalized to any pair whose greatest common divisor is greater than 1.

1. Used for deciding whether a modular inverse exists.

2. In this clip the rule is stated for the case gcd > 1 implying no inverse.

1. Meaning of a modular inverse in this lesson

1. Formula
   Observation

   The theorem statement is visible on screen: “The integer a has an inverse modulo m if and only if (a,m)=1.”

2. Audio
   Observation

   The speaker reads the theorem aloud and explains the case gcd(a,m)>1 means no inverse exists.

For integers a and m, a has a multiplicative inverse modulo m if and only if (a,m)=1.

1. a is an integer.

2. m is the modulus.

For the integers a and m appearing in the statement.

1. Audio
   Observation

   The speaker says, “If the greatest common divisor between a and m is greater than 1, then there is no inverse mod m.”

If the greatest common divisor of a and m is greater than 1, then a has no inverse modulo m.

1. a and m are integers.

2. (a,m)>1.

For integers a and m satisfying the gcd condition.

1. Formula
   Observation

   The board writes $18^{-1}$ ≡ 18 after showing $324 = 18\cdot 18$, and again after using 18 ≡ -1 (mod 19).

2. Audio
   Observation

   The speaker states, 'So 18's inverse is 18' and later '18 is its own inverse.'

$18^{-1}$ ≡ 18 (mod 19).

1. Work modulo 19.

2. Use the fact that 18 ≡ -1 (mod 19), or equivalently that $18\cdot 18 = 324$ ≡ 1 (mod 19).

Specific numerical statement for modulus 19.

1. Formula
   Observation

   The board writes 17 ≡ -2 (mod 19), then (-2)^{-1} ≡ -10 (mod 19), then $17^{-1}$ ≡ 9 (mod 19).

2. Audio
   Observation

   The speaker concludes that the inverse of 17 is congruent to 9 mod 19.

$17^{-1}$ ≡ 9 (mod 19).

1. Work modulo 19.

2. Use 17 ≡ -2 (mod 19) and $20 = (-2)(-10)$, so (-2)^{-1} ≡ -10 (mod 19).

3. Convert -10 to the positive representative 9 modulo 19.

Specific numerical statement for modulus 19.

1. Audio
   Observation

   The speaker asks what multiplied by negative 1 gives positive 1 and answers negative 1.

2. Formula
   Observation

   The reasoning is written through 18 ≡ -1 (mod 19) and the conclusion $18^{-1}$ ≡ 18 (mod 19).

(-1)^{-1} ≡ -1 (mod m), because (-1)(-1)=1.

1. Any modulus m is allowed for this general fact.

2. In the clip it is applied with $m=19$.

General proposition applied to modulus 19 in the example.

1. Audio
   Observation

   "So if there's a greatest common divisor greater than 1 between those two values, then there is no inverse."

2. Formula
   Observation

   Board text: (5,10)=5 NOT 1, followed by No inverse.

1. The clip states only the direction gcd>1 implies no inverse; it does not explicitly state the converse condition gcd=1 implies existence.

For the numbers under consideration, if the greatest common divisor of a and m is greater than 1, then a has no modular inverse modulo m.

1. a and m are integers with modulus m.

2. (a,m)>1.

The spoken wording is general over "those two values," but the clip only proves or illustrates the claim in the specific example $a=5$, $m=10$.

1. Formula
   Observation

   Board text includes $17^{-1} \equiv 9 \pmod{19}$.

2. Audio
   Observation

   The speaker says that eventually one would get to 17 times 9, which would be 153.

$17^{-1} \equiv 9 \pmod{19}$.

1. Working modulo 19.

2. Using the intermediate facts $17 \equiv -2 \pmod{19}$ and (-2)^{-1} $\equiv -10 \pmod{19}$ shown on the board.

Specific numerical claim.

1. Formula
   Observation

   Board text includes $18 \equiv -1 \pmod{19}$ and $18^{-1} \equiv 1 \pmod{19}$.

$18^{-1} \equiv 1 \pmod{19}$.

1. Working modulo 19.

2. Using $18 \equiv -1 \pmod{19}$.

Specific numerical claim.

1. Formula
   Observation

   The board shows the chain “1 ≡ 8 ≡ 15 (mod 7)” with 15 factored below into 5 and 3.

2. Audio
   Observation

   The speaker says, “Mod 7, 1 is congruent to 8. That's not a multiple of 5. So we add 7 again and we get 15. 15 is 5 times 3 mod 7.”

3. Formula
   Observation

   The final written conclusion is “$5^{-1}$ ≡ 3 (mod 7)”.

1. Expression

   $$1 \equiv 8 \pmod{7}$$
   Explanation

   Start from 1 and add the modulus 7 once to obtain another representative congruent to 1 modulo 7.

   Justification

   Adding the modulus preserves congruence modulo 7.

   Shown in the video
2. Expression

   $$8 \not\equiv 0 \pmod{5}$$
   Explanation

   Check whether 8 is a multiple of 5; it is not, so continue searching.

   Justification

   The method requires finding a representative congruent to 1 modulo 7 that is divisible by 5.

   Derived from the video
3. Expression

   $$1 \equiv 8 \equiv 15 \pmod{7}$$
   Explanation

   Add 7 again to reach 15, which is still congruent to 1 modulo 7.

   Justification

   Repeated addition of the modulus stays within the same congruence class.

   Shown in the video
4. Expression

   $$15 = 5 \cdot 3$$
   Explanation

   Factor 15 as 5 times 3.

   Justification

   Ordinary integer factorization.

   Shown in the video
5. Expression

   $$5^{-1} \equiv 3 \pmod{7}$$
   Explanation

   Since $5\cdot 3$ ≡ 15 ≡ 1 (mod 7), the inverse of 5 modulo 7 is 3.

   Justification

   By the definition of multiplicative inverse modulo m.

   Shown in the video

The inverse of 5 modulo 7 is 3, written as $5^{-1}$ ≡ 3 (mod 7).

1. Formula
   Observation

   The board shows “1 ≡ 12 ≡ 23 ≡ 34 ≡ 45 (mod 11)” with 45 factored into 5 and 9.

2. Audio
   Observation

   The speaker says, “So hopefully you said 1 was congruent to 12 mod 11. Always start with 1 and add the mod on. Congruent to 23, adding 11, 34, and eventually we get to 45. And that's 5 times 9 mod 11.”

3. Formula
   Observation

   The final written conclusion is “$5^{-1}$ ≡ 9 (mod 11)”.

1. Expression

   $$1 \equiv 12 \pmod{11}$$
   Explanation

   Begin with 1 and add the modulus 11 to get 12.

   Justification

   Adding the modulus preserves congruence modulo 11.

   Shown in the video
2. Expression

   $$12 \equiv 23 \equiv 34 \equiv 45 \pmod{11}$$
   Explanation

   Continue adding 11 until reaching a number divisible by 5.

   Justification

   All these numbers remain congruent to 1 modulo 11.

   Shown in the video
3. Expression

   $$45 = 5 \cdot 9$$
   Explanation

   Factor 45 as 5 times 9.

   Justification

   Ordinary integer factorization.

   Shown in the video
4. Expression

   $$5^{-1} \equiv 9 \pmod{11}$$
   Explanation

   Because $5\cdot 9 = 45$ ≡ 1 (mod 11), the inverse of 5 modulo 11 is 9.

   Justification

   By the definition of multiplicative inverse modulo m.

   Shown in the video

The inverse of 5 modulo 11 is 9, written as $5^{-1}$ ≡ 9 (mod 11).

1. Formula
   Observation

   The board shows “1 ≡ 18 (mod 17)” with 18 factored into 6 and 3.

2. Audio
   Observation

   The speaker says, “So you've got 1 congruent to 18 because we're adding 17 this time, and that's 6 times 3 mod 17.”

3. Formula
   Observation

   The final written conclusion is “$6^{-1}$ ≡ 3 (mod 17)”.

1. Expression

   $$1 \equiv 18 \pmod{17}$$
   Explanation

   Start from 1 and add the modulus 17 to obtain 18.

   Justification

   Adding the modulus preserves congruence modulo 17.

   Shown in the video
2. Expression

   $$18 = 6 \cdot 3$$
   Explanation

   Factor 18 as 6 times 3.

   Justification

   Ordinary integer factorization.

   Shown in the video
3. Expression

   $$6^{-1} \equiv 3 \pmod{17}$$
   Explanation

   Since $6\cdot 3 = 18$ ≡ 1 (mod 17), the inverse of 6 modulo 17 is 3.

   Justification

   By the definition of multiplicative inverse modulo m.

   Shown in the video

The inverse of 6 modulo 17 is 3, written as $6^{-1}$ ≡ 3 (mod 17).

1. Formula
   Observation

   The board lists 1 ≡ 20 ≡ 39 ≡ 58 ≡ 77 ≡ ... ≡ 324 and factors 324 as $18\cdot 18$.

2. Audio
   Observation

   The speaker narrates adding 19 repeatedly until reaching 324, which is 18 times 18.

1. The intermediate omitted terms between 77 and 324 are not all written explicitly; only the ellipsis is shown.

1. Expression

   $$1 \equiv 20 \equiv 39 \equiv 58 \equiv 77 \equiv \cdots \equiv 324 \pmod{19}$$
   Explanation

   Start from 1 and keep adding the modulus 19 to generate numbers congruent to 1 modulo 19.

   Justification

   If x ≡ 1 (mod 19), then $x+19$ ≡ 1 (mod 19).

   Shown in the video
2. Expression

   $$324 = 18 \cdot 18$$
   Explanation

   Identify the first listed number that is a multiple of 18.

   Justification

   Direct factorization shown on the board.

   Shown in the video
3. Expression

   $$18^{-1} \equiv 18 \pmod{19}$$
   Explanation

   Since $18\cdot 18$ ≡ 1 (mod 19), the inverse of 18 modulo 19 is 18.

   Justification

   Definition of modular multiplicative inverse.

   Shown in the video

The brute-force search proves that $18^{-1}$ ≡ 18 (mod 19).

1. Formula
   Observation

   The board writes 18 ≡ -1 (mod 19) and then $18^{-1}$ ≡ 18 (mod 19).

2. Audio
   Observation

   The speaker explains that instead of adding 19, we can subtract 19, and that the inverse of -1 is -1.

1. Expression

   $$18 \equiv -1 \pmod{19}$$
   Explanation

   Replace 18 by the congruent negative residue -1.

   Justification

   Subtracting the modulus preserves congruence: $18-19=-1$.

   Shown in the video
2. Expression

   $$(-1)(-1)=1$$
   Explanation

   Recognize that -1 multiplied by itself gives 1.

   Justification

   Basic integer multiplication.

   Shown in the video
3. Expression

   $$18^{-1} \equiv 18 \pmod{19}$$
   Explanation

   Therefore the inverse of 18 is itself modulo 19.

   Justification

   Because 18 ≡ -1 and (-1)^{-1} ≡ -1.

   Shown in the video

Using the negative residue -1 gives a much faster proof that 18 is its own inverse modulo 19.

1. Formula
   Observation

   The board writes 17 ≡ -2 (mod 19), notes $20 = (-2)(-10)$, writes (-2)^{-1} ≡ -10 (mod 19), and finally $17^{-1}$ ≡ 9 (mod 19).

2. Audio
   Observation

   The speaker says 17 is the same thing as negative 2 mod 19, looks back at 20 = negative 2 times negative 10, and then adds 19 to negative 10 to get a positive number.

1. Expression

   $$17 \equiv -2 \pmod{19}$$
   Explanation

   Rewrite 17 as the negative residue -2.

   Justification

   Adding 19 to -2 gives 17, so they are congruent modulo 19.

   Shown in the video
2. Expression

   $$20 = (-2)(-10)$$
   Explanation

   Use the earlier line 1 ≡ 20 (mod 19) and factor 20 as (-2)(-10).

   Justification

   Direct arithmetic identity shown on the board.

   Shown in the video
3. Expression

   $$(-2)^{-1} \equiv -10 \pmod{19}$$
   Explanation

   Since (-2)(-10)=20 ≡ 1 (mod 19), -10 is the inverse of -2 modulo 19.

   Justification

   Definition of modular inverse together with 20 ≡ 1 (mod 19).

   Shown in the video
4. Expression

   $$-10 + 19 = 9$$
   Explanation

   Convert the negative representative -10 to the positive residue 9.

   Justification

   Adding the modulus does not change the congruence class.

   Shown in the video
5. Expression

   $$17^{-1} \equiv 9 \pmod{19}$$
   Explanation

   Therefore the inverse of 17 modulo 19 is 9.

   Justification

   Substitute 17 ≡ -2 and the converted inverse.

   Shown in the video

The shortcut method yields $17^{-1}$ ≡ 9 (mod 19).

1. Formula
   Observation

   The board shows $18 \equiv -1 \pmod{19}$ and directly below it $18^{-1} \equiv 1 \pmod{19}$.

1. Expression

   $$18 \equiv -1 \pmod{19}$$
   Explanation

   Rewrite 18 as its negative residue modulo 19.

   Justification

   Congruence reduction modulo 19 shown on the board.

   Shown in the video
2. Expression

   $$18^{-1} \equiv 1 \pmod{19}$$
   Explanation

   Conclude that the inverse of 18 modulo 19 is 1.

   Justification

   The video presents this as the immediate consequence of the preceding congruence; the underlying fact is that -1 is its own inverse modulo 19.

   Shown in the video

$18^{-1} \equiv 1 \pmod{19}$.

1. Formula
   Observation

   The board shows $17 \equiv -2 \pmod{19}$, (-2)^{-1} $\equiv -10 \pmod{19}$, and $17^{-1} \equiv 9 \pmod{19}$.

2. Audio
   Observation

   The speaker mentions the longer route ending at $17 \times 9 = 153$.

1. The clip does not show the intermediate verification $153 \equiv 1 \pmod{19}$; it only states $17 \times 9 = 153$ in the audio.

1. Expression

   $$17 \equiv -2 \pmod{19}$$
   Explanation

   Replace 17 by the smaller negative residue -2 modulo 19.

   Justification

   Congruence reduction modulo 19 shown on the board.

   Shown in the video
2. Expression

   $$(-2)^{-1} \equiv -10 \pmod{19}$$
   Explanation

   Use the inverse of -2 modulo 19.

   Justification

   Written directly on the board as a known intermediate step.

   Shown in the video
3. Expression

   $$17^{-1} \equiv 9 \pmod{19}$$
   Explanation

   Convert -10 to the positive residue 9 modulo 19 and record the inverse of 17.

   Justification

   Since -10 $\equiv 9 \pmod{19}$, the board rewrites the inverse in positive form.

   Shown in the video

$17^{-1} \equiv 9 \pmod{19}$.

1. Formula
   Observation

   The board writes $1 \equiv 11 \equiv 21 \equiv 31 \equiv$ and then (5,10)=5 NOT 1 and No inverse.

2. Audio
   Observation

   The speaker says adding 10 to 1 gives 11, 21, 31, and one would never get a multiple of 5.

3. Audio
   Observation

   The speaker identifies the problem as gcd(5,10)=5, not 1.

1. The written congruence line is visually incomplete after the final $\equiv$ sign; the intended continuation is supplied by the audio as 11, 21, 31, ...

1. Expression

   $$1 \equiv 11 \equiv 21 \equiv 31 \equiv \cdots \pmod{10}$$
   Explanation

   Begin listing numbers congruent to 1 modulo 10 by repeatedly adding 10.

   Justification

   The board shows $1 \equiv 11 \equiv 21 \equiv 31 \equiv$, and the audio explains the pattern of adding 10.

   Shown in the video
2. Expression

   $$\text{No term in }1,11,21,31,\ldots\text{ is a multiple of }5$$
   Explanation

   Observe that this sequence never reaches a multiple of 5.

   Justification

   Stated explicitly in the audio: "you would never get a multiple of 5."

   Shown in the video
3. Expression

   $$(5,10)=5 \ne 1$$
   Explanation

   Compute the greatest common divisor of 5 and 10.

   Justification

   Written on the board and spoken aloud.

   Shown in the video
4. Expression

   $$\text{No inverse}$$
   Explanation

   Conclude that 5 has no inverse modulo 10.

   Justification

   The speaker states the general rule that gcd greater than 1 implies no inverse.

   Shown in the video

There is no inverse of 5 modulo 10 because (5,10)=$5 \ne 1$.

1. Formula
   Observation

   Prompt on screen: “Find the inverse of 5 mod 7.”

2. Formula
   Observation

   Work shown: “1 ≡ 8 ≡ 15 (mod 7)” with 15 split into 5 and 3.

3. Formula
   Observation

   Conclusion written: “$5^{-1}$ ≡ 3 (mod 7)”.

Find the inverse of 5 mod 7.

1. Target integer: 5.

2. Modulus: 7.

Find an integer b such that 5b ≡ 1 (mod 7).

1. Expression

   $$1 \equiv 8 \pmod{7}$$
   Explanation

   Add 7 to 1 to search within the congruence class of 1 modulo 7.

   Justification

   Congruence is preserved when adding the modulus.

   Shown in the video
2. Expression

   $$1 \equiv 8 \equiv 15 \pmod{7}$$
   Explanation

   Add 7 again to reach 15.

   Justification

   15 is still congruent to 1 modulo 7.

   Shown in the video
3. Expression

   $$15 = 5 \cdot 3$$
   Explanation

   Recognize 15 as a multiple of 5.

   Justification

   Integer factorization.

   Shown in the video
4. Expression

   $$5^{-1} \equiv 3 \pmod{7}$$
   Explanation

   Conclude that 3 is the inverse of 5 modulo 7.

   Justification

   Because $5\cdot 3$ ≡ 1 (mod 7).

   Shown in the video

$5^{-1} \equiv 3 \pmod{7}$

$5\cdot 3 = 15$ and 15 ≡ 1 (mod 7).

1. Formula
   Observation

   Prompt on screen: “Find the inverse of 5 mod 11.”

2. Formula
   Observation

   Work shown: “1 ≡ 12 ≡ 23 ≡ 34 ≡ 45 (mod 11)” with 45 split into 5 and 9.

3. Formula
   Observation

   Conclusion written: “$5^{-1}$ ≡ 9 (mod 11)”.

Find the inverse of 5 mod 11.

1. Target integer: 5.

2. Modulus: 11.

Find an integer b such that 5b ≡ 1 (mod 11).

1. Expression

   $$1 \equiv 12 \pmod{11}$$
   Explanation

   Start with 1 and add 11.

   Justification

   Adding the modulus preserves congruence modulo 11.

   Shown in the video
2. Expression

   $$12 \equiv 23 \equiv 34 \equiv 45 \pmod{11}$$
   Explanation

   Keep adding 11 until reaching a multiple of 5.

   Justification

   All listed numbers remain congruent to 1 modulo 11.

   Shown in the video
3. Expression

   $$45 = 5 \cdot 9$$
   Explanation

   Factor 45 as 5 times 9.

   Justification

   Integer factorization.

   Shown in the video
4. Expression

   $$5^{-1} \equiv 9 \pmod{11}$$
   Explanation

   Conclude that 9 is the inverse of 5 modulo 11.

   Justification

   Because $5\cdot 9$ ≡ 1 (mod 11).

   Shown in the video

$5^{-1} \equiv 9 \pmod{11}$

$5\cdot 9 = 45$ and 45 ≡ 1 (mod 11).

1. Formula
   Observation

   Prompt on screen: “Find the inverse of 6 mod 17.”

2. Formula
   Observation

   Work shown: “1 ≡ 18 (mod 17)” with 18 split into 6 and 3.

3. Formula
   Observation

   Conclusion written: “$6^{-1}$ ≡ 3 (mod 17)”.

Find the inverse of 6 mod 17.

1. Target integer: 6.

2. Modulus: 17.

Find an integer b such that 6b ≡ 1 (mod 17).

1. Expression

   $$1 \equiv 18 \pmod{17}$$
   Explanation

   Add 17 to 1 to get 18.

   Justification

   Adding the modulus preserves congruence modulo 17.

   Shown in the video
2. Expression

   $$18 = 6 \cdot 3$$
   Explanation

   Factor 18 as 6 times 3.

   Justification

   Integer factorization.

   Shown in the video
3. Expression

   $$6^{-1} \equiv 3 \pmod{17}$$
   Explanation

   Conclude that 3 is the inverse of 6 modulo 17.

   Justification

   Because $6\cdot 3$ ≡ 1 (mod 17).

   Shown in the video

$6^{-1} \equiv 3 \pmod{17}$

$6\cdot 3 = 18$ and 18 ≡ 1 (mod 17).

1. Caption evidence
   Observation

   The typed prompt on screen reads 'Find the inverse of 18 mod 19.'

2. Formula
   Observation

   The board solves it first by listing 1 ≡ 20 ≡ 39 ≡ 58 ≡ 77 ≡ ... ≡ 324 and then by writing 18 ≡ -1 (mod 19).

Find the inverse of 18 mod 19.

1. Modulus $m = 19$.

2. Residue $a = 18$.

Find b such that 18b ≡ 1 (mod 19).

1. Expression

   $$1 \equiv 20 \equiv 39 \equiv 58 \equiv 77 \equiv \cdots \equiv 324 \pmod{19}$$
   Explanation

   Generate numbers congruent to 1 modulo 19 by repeatedly adding 19.

   Justification

   Congruence is preserved when adding the modulus.

   Shown in the video
2. Expression

   $$324 = 18 \cdot 18$$
   Explanation

   Notice that 324 is divisible by 18, with quotient 18.

   Justification

   Factorization shown on the board.

   Shown in the video
3. Expression

   $$18^{-1} \equiv 18 \pmod{19}$$
   Explanation

   Conclude that 18 is its own inverse modulo 19.

   Justification

   Because $18\cdot 18$ ≡ 1 (mod 19).

   Shown in the video
4. Expression

   $$18 \equiv -1 \pmod{19},\quad (-1)^2=1$$
   Explanation

   A faster route is to rewrite 18 as -1 and use that -1 is self-inverse.

   Justification

   Negative residues are congruent modulo 19, and (-1)(-1)=1.

   Shown in the video

$18^{-1}$ ≡ 18 (mod 19).

Check: $18\cdot 18 = 324$ and 324 ≡ 1 (mod 19); equivalently, 18 ≡ -1 and (-1)^2 ≡ 1.

1. Caption evidence
   Observation

   The handwritten prompt reads 'Find inverse of 17 mod 19.'

2. Formula
   Observation

   The board writes 17 ≡ -2 (mod 19), (-2)^{-1} ≡ -10 (mod 19), and $17^{-1}$ ≡ 9 (mod 19).

Find the inverse of 17 mod 19.

1. Modulus $m = 19$.

2. Residue $a = 17$.

3. Earlier in the clip, 1 ≡ 20 (mod 19) was already established.

Find b such that 17b ≡ 1 (mod 19).

1. Expression

   $$17 \equiv -2 \pmod{19}$$
   Explanation

   Replace 17 by the simpler negative residue -2.

   Justification

   Adding 19 to -2 gives 17.

   Shown in the video
2. Expression

   $$20 = (-2)(-10)$$
   Explanation

   Factor the previously noted number 20, which is congruent to 1 modulo 19.

   Justification

   Direct arithmetic identity.

   Shown in the video
3. Expression

   $$(-2)^{-1} \equiv -10 \pmod{19}$$
   Explanation

   Thus -10 is the inverse of -2 modulo 19.

   Justification

   Because (-2)(-10)=20 ≡ 1 (mod 19).

   Shown in the video
4. Expression

   $$-10 + 19 = 9$$
   Explanation

   Convert the negative inverse to a positive representative.

   Justification

   Adding the modulus preserves congruence.

   Shown in the video
5. Expression

   $$17^{-1} \equiv 9 \pmod{19}$$
   Explanation

   Therefore the inverse of 17 modulo 19 is 9.

   Justification

   Substitute back 17 ≡ -2.

   Shown in the video

$17^{-1}$ ≡ 9 (mod 19).

Check: $17\cdot 9 = 153$ and $153 = 8\cdot 19 + 1$, so $17\cdot 9$ ≡ 1 (mod 19).

1. Formula
   Observation

   Printed prompt: "Find the inverse of 18 mod 19."

2. Formula
   Observation

   Handwritten work includes $18 \equiv -1 \pmod{19}$, $18^{-1} \equiv 1 \pmod{19}$, Find inverse of 17 mod 19, $17 \equiv -2 \pmod{19}$, (-2)^{-1} $\equiv -10 \pmod{19}$, $17^{-1} \equiv 9 \pmod{19}$.

3. Audio
   Observation

   The speaker says the long way would reach $17 \times 9 = 153$.

1. The top congruence chain visible on the board is only partially legible in this clip; the exact full sequence before 324 cannot be transcribed reliably from the provided frames.

Compute the modular inverses of 18 and 17 modulo 19.

1. Modulus $m=19$.

2. Numbers whose inverses are requested: 18 and 17.

3. Intermediate congruences shown: $18 \equiv -1 \pmod{19}$, $17 \equiv -2 \pmod{19}$, (-2)^{-1} $\equiv -10 \pmod{19}$.

Express each inverse as a residue modulo 19.

1. Expression

   $$18 \equiv -1 \pmod{19}$$
   Explanation

   Reduce 18 to a simpler representative modulo 19.

   Justification

   Shown on the board.

   Shown in the video
2. Expression

   $$18^{-1} \equiv 1 \pmod{19}$$
   Explanation

   Record the inverse of 18.

   Justification

   Directly written on the board as the next line.

   Shown in the video
3. Expression

   $$17 \equiv -2 \pmod{19}$$
   Explanation

   Reduce 17 to a simpler representative modulo 19.

   Justification

   Shown on the board under "Find inverse of 17 mod 19."

   Shown in the video
4. Expression

   $$(-2)^{-1} \equiv -10 \pmod{19}$$
   Explanation

   Use the inverse of -2 modulo 19.

   Justification

   Shown on the board.

   Shown in the video
5. Expression

   $$17^{-1} \equiv 9 \pmod{19}$$
   Explanation

   Rewrite -10 as 9 modulo 19 to give the positive residue for the inverse of 17.

   Justification

   Shown on the board; the conversion uses -10 $\equiv 9 \pmod{19}$.

   Shown in the video

$18^{-1} \equiv 1 \pmod{19}$ and $17^{-1} \equiv 9 \pmod{19}$.

The audio notes that the longer route for 17 reaches $17 \times 9 = 153$, consistent with checking the product before reducing modulo 19.

1. Formula
   Observation

   Printed prompt: "Find the inverse of 5 mod 10."

2. Formula
   Observation

   Handwritten work includes $1 \equiv 11 \equiv 21 \equiv 31 \equiv$, (5,10)=5 NOT 1, and No inverse.

3. Audio
   Observation

   The speaker says adding 10 repeatedly never yields a multiple of 5, then states gcd(5,10)=5, not 1, so there is no inverse.

1. The written congruence line ends with a trailing $\equiv$ sign without a visible next numeral; the audio supplies the intended continuing pattern.

Determine whether 5 has a multiplicative inverse modulo 10, and find it if it exists.

1. Modulus $m=10$.

2. Number whose inverse is requested: 5.

Decide existence of $5^{-1} \pmod{10}$ and conclude accordingly.

1. Expression

   $$1 \equiv 11 \equiv 21 \equiv 31 \equiv \cdots \pmod{10}$$
   Explanation

   Try the direct search by listing numbers congruent to 1 modulo 10.

   Justification

   The board writes the beginning of this chain, and the audio explains adding 10 to 1, then 11, 21, 31.

   Shown in the video
2. Expression

   $$\text{never get a multiple of }5$$
   Explanation

   Notice that none of these candidates is divisible by 5, so no product 5x can equal 1 modulo 10 by this route.

   Justification

   Explicitly stated in the audio.

   Shown in the video
3. Expression

   $$(5,10)=5$$
   Explanation

   Compute the greatest common divisor of 5 and 10.

   Justification

   Written on the board and spoken aloud.

   Shown in the video
4. Expression

   $$5 \ne 1$$
   Explanation

   Compare the gcd with 1.

   Justification

   The board writes NOT 1 beside (5,10)=5.

   Shown in the video
5. Expression

   $$\text{No inverse}$$
   Explanation

   Conclude that 5 has no inverse modulo 10.

   Justification

   Written on the board and justified by the spoken gcd rule.

   Shown in the video

No inverse.

The example is checked by the explicit gcd computation (5,10)=$5 \ne 1$ and by the failed search through 1, 11, 21, 31, ...

1. Diagram
   Observation

   A white slide titled “Modular Inverses” shows a definition line and a theorem line in black text.

2. Animation
   Observation

   Purple underlines are drawn sequentially beneath key phrases and symbols in the definition and theorem.

1. Title “Modular Inverses”.

2. Definition sentence.

3. Theorem sentence.

4. Purple underline strokes.

1. Underline appears under “multiplicative inverses modulo m” around 10s.

2. Underline appears under “ab ≡ 1 (mod m)” around 12s–14s.

3. Underline appears under “a” in the theorem around 25s.

4. Underline appears under “modulo m” around 26s–27s.

5. Underline appears under “(a,m)=1” around 28s–29s.

1. The slide text itself remains unchanged while annotations are added.

The visual emphasis tracks the spoken explanation, highlighting the definition of inverse and the gcd condition in the theorem.

1. Diagram
   Observation

   New slide prompt: “Find the inverse of 5 mod 7.”

2. Animation
   Observation

   Purple handwriting builds the congruence chain and factor tree step by step.

1. Prompt text.

2. Handwritten congruence chain “1 ≡ 8 ≡ 15 (mod 7)”.

3. Factor branches from 15 to 5 and 3.

4. Final notation “$5^{-1}$ ≡ 3 (mod 7)”.

1. First “1 ≡ 8” is written.

2. Then “≡ 15” is appended.

3. Then 15 is factored into 5 and 3 below.

4. Finally the inverse statement is written underneath.

1. The modulus remains 7 throughout the example.

The animation shows the search process: move through numbers congruent to 1 modulo 7 until a multiple of 5 is found, then read off the inverse.

1. Diagram
   Observation

   New slide prompt: “Find the inverse of 5 mod 11.”

2. Animation
   Observation

   Purple handwriting extends a longer congruence chain and then factors the terminal number.

1. Prompt text.

2. Handwritten chain “1 ≡ 12 ≡ 23 ≡ 34 ≡ 45 (mod 11)”.

3. Factor branches from 45 to 5 and 9.

4. Final notation “$5^{-1}$ ≡ 9 (mod 11)”.

1. The chain grows term by term from 12 to 23 to 34 to 45.

2. After 45 is reached, it is factored into 5 and 9.

3. The inverse statement is then written below.

1. The modulus remains 11 throughout the example.

This visual sequence demonstrates the same add-the-modulus search method with more iterations before hitting a multiple of 5.

1. Diagram
   Observation

   New slide prompt: “Find the inverse of 6 mod 17.”

2. Animation
   Observation

   Purple handwriting shows “1 ≡ 18 (mod 17)” and factors 18 into 6 and 3.

1. Prompt text.

2. Handwritten congruence “1 ≡ 18 (mod 17)”.

3. Factor branches from 18 to 6 and 3.

4. Final notation “$6^{-1}$ ≡ 3 (mod 17)”.

1. 18 is written as the next representative after 1.

2. 18 is factored into 6 and 3.

3. The inverse statement is written below.

1. The modulus remains 17 throughout the example.

The shortest example shows that sometimes one addition of the modulus already yields a multiple of the target integer.

1. Animation
   Observation

   The screen changes from a previous problem about 6 mod 17 to a new typed prompt 'Find the inverse of 18 mod 19.'

1. Previous handwritten solution for 6 mod 17

2. New typed prompt for 18 mod 19

1. Old work disappears.

2. New problem statement appears at the top.

1. The interface remains a digital whiteboard with purple handwriting.

This marks the start of the main example in the clip.

1. Animation
   Observation

   Purple handwriting successively adds terms to the chain 1 ≡ 20 ≡ 39 ≡ 58 ≡ 77 ≡ ... ≡ 324, then writes the factor pair 18 and 18 beneath 324.

1. Congruence chain beginning with 1

2. Number 324

3. Factor labels 18 and 18

1. Each new term is obtained by adding 19.

2. The chain ends at 324.

3. 324 is decomposed visually into $18\cdot 18$.

1. All displayed terms remain in the same congruence class as 1 modulo 19.

The visual process demonstrates a search for a multiple of 18 inside the congruence class of 1 modulo 19.

1. Animation
   Observation

   Below the earlier work, the presenter writes 18 ≡ -1 (mod 19), then rewrites the inverse conclusion as $18^{-1}$ ≡ 18 (mod 19).

1. Expression 18 ≡ -1 (mod 19)

2. Expression $18^{-1}$ ≡ 18 (mod 19)

1. The positive residue 18 is replaced by the negative residue -1.

2. The inverse statement is restated using the shortcut.

1. The modulus stays 19.

2. The residue class represented by 18 does not change.

The visual rewrite emphasizes that choosing a convenient representative can simplify inverse calculations.

1. Animation
   Observation

   A horizontal divider is drawn, then the new prompt 'Find inverse of 17 mod 19' is written, followed by 17 ≡ -2 (mod 19), (-2)^{-1} ≡ -10 (mod 19), and $17^{-1}$ ≡ 9 (mod 19).