How do you start the Euclidean algorithm for gcd(10,45)?
Conditions
- The inputs are positive integers.
- The larger number is placed on the left-hand side of the equation.
- The quotient is an integer and the remainder satisfies .
Reasoning, step by step
- Identify the two integers, 10 and 45.
- Place the larger number (45) on the left side of the equation.
- Set the larger number equal to the smaller number (10) multiplied by an unknown quotient plus an unknown remainder : .
- Determine how many whole times 10 fits into 45, which gives the quotient .
- Calculate the leftover amount after subtracting 4 copies of 10 from 45, which gives the remainder .
- Substitute the values back into the equation to get .
Example
The board writes "" and then "". The speaker explains that q is how many times 10 goes into 45, and r is the remainder of that result.
Common misconceptions
- Assuming the smaller number should be on the left side of the equation.
- Forgetting that the remainder must be strictly less than the divisor.
- Confusing the quotient with the remainder.
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Conditions: The algorithm is applied to two positive integers.; The previous remainder is not zero.; The process continues until a remainder of 0 is obtained.
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In the Euclidean algorithm, the previous divisor becomes the new dividend (placed on the left side of the equation), and the previous remainder becomes the new divisor (placed on the right side). This recursive shift carries the numbers forward so that each step divides the former divisor by the former remainder, continuing until a remainder of zero is reached.
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