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The inverse of 17 modulo 19 is 9. It is computed quickly by rewriting 17 as -2 modulo 19.

Conditions: Modulus is 19.; Residue $a = 17$.

To calculate the probability of rolling an even number three times, first find the probability of rolling an even number on a single roll, which is $1/2$ for a fair six-sided die. Then, assuming the three rolls are mutually independent, multiply this single-roll probability by itself three times: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.

Conditions: The die is fair (six-sided, equally likely outcomes).; The three rolls are mutually independent.; The target event is rolling an even number on every one of the three rolls.

The system Ax=b is consistent exactly when the vector b lies in the column space of A. Matrix dimensions restrict the possible rank, but the shape of the matrix alone does not guarantee consistency or uniqueness.

Conditions: A is a matrix representing a linear transformation.; b is a vector with compatible dimensions.

A sigma-algebra on a set $\Omega$ is a collection $\mathcal{F}$ of subsets of $\Omega$ that satisfies three specific closure properties: it must contain the whole space $\Omega$, it must be closed under complements, and it must be closed under countable unions.

Conditions: $\Omega$ is a set.; $\mathcal{F}$ is a collection of subsets of $\Omega$.

In this video, the Euclidean algorithm is introduced as a method for finding the greatest common factor of two numbers. The presentation specifically demonstrates the subtraction version of the algorithm rather than the modulo version.

Conditions: Applies to two numbers in the worked example.; The video uses positive integer examples.

In the job-sequencing state-space tree, a root-to-node path records the ordered partial sequence of jobs selected so far. Each edge label appends one job to the current sequence, so following a route (e.g., $J_1$ then $J_4$) means those jobs are chosen in that order while jobs not on the path are excluded from this partial sequence. This interpretation applies specifically to the drawn tree where edges represent incremental job additions, and it clarifies that a partial path is not a complete schedule.

Conditions: The tree is the job-sequencing state-space tree drawn on the board.; Each edge label indicates appending one job to the current partial sequence.

The author of the textbook is Professor Wang Gaoxiong. The instructor identifies him verbally and also writes the name by hand on the screen for emphasis.

Conditions: Specific to the textbook 'Ordinary Differential Equations' used in this course.

Naturality in $A$ means that the adjunction correspondence is compatible with precomposition by any morphism $h:A'\to A$. Specifically, taking a morphism $A\to G(B)$, precomposing with $h$, and then translating across the adjunction gives the same result as first translating $A\to G(B)$ to $F(A)\to B$ and then precomposing with $F(h):F(A')\to F(A)$.

Conditions: Fix an object $B\in\mathcal{B}$.; The statement quantifies over all morphisms $h:A'\to A$ in $\mathcal{A}$.; The correspondence is between morphisms in $\mathcal{A}$ and morphisms in $\mathcal{B}$.

To compute $\gcd(10,15)$, repeatedly apply the identity $\gcd(a,b)=\gcd(b,a\bmod b)$ until the second argument becomes zero. Starting from $\gcd(10,15)$, the sequence proceeds as $\gcd(15,10)$, then $\gcd(10,5)$, and finally $\gcd(5,0)$. Since the second argument is zero, the algorithm terminates and returns the first argument, 5, as the greatest common divisor.

Conditions: The initial pair is (10, 15).; The algorithm uses the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$.; The process stops when the second argument is zero.

The clock example illustrates wrap-around by showing that although a day has 24 hours, a 12-hour clock uses only 12 numbers. Times beyond 12 map back onto the same cycle, such as 15 hours becoming 3 p.m.

Conditions: The example uses a standard 12-hour analog clock.; The modulus is 12.

Piecewise functions are defined by different formulas on different intervals. To determine the limit as $x$ approaches a boundary point $a$, you must use the formula valid for $x < a$ for the left-hand limit and the formula valid for $x > a$ for the right-hand limit.

Conditions: The function is defined piecewise.; The point $a$ is a boundary between branches.

Scheme theory enlarges the notion of an algebraic variety by taking into account multiplicities and allowing varieties to be defined over any commutative ring, not just algebraically closed fields. For example, the equations $x = 0$ and $x^2 = 0$ define the same algebraic variety but different schemes.

Conditions: The context is the motivation for scheme theory over classical algebraic geometry.; Schemes account for multiplicities.; Schemes can be defined over any commutative ring.