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In the step $45 = 10 \cdot q + r$, $q$ represents the quotient, which counts how many whole times the smaller number (10) fits into the larger number (45). $r$ represents the remainder, which is the leftover amount after subtracting those whole multiples.

Conditions: The equation is part of the division-with-remainder step of the Euclidean algorithm.; The inputs are positive integers.; The remainder satisfies $0 \le r < 10$.

In the step $45 = 10 \cdot q + r$, $q$ represents the quotient, which counts how many whole times the smaller number (10) fits into the larger number (45). $r$ represents the remainder, which is the leftover amount after subtracting those whole multiples.

Conditions: The equation is part of the division-with-remainder step of the Euclidean algorithm.; The inputs are positive integers.; The remainder satisfies $0 \le r < 10$.

To start the Euclidean algorithm for $\gcd(10,45)$, you write the larger number as the smaller number multiplied by an unknown quotient plus an unknown remainder. Specifically, you set up the division equation $45 = 10 \cdot q + r$.

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 $0 \le r < 10$.

To start the Euclidean algorithm for $\gcd(10,45)$, you write the larger number as the smaller number multiplied by an unknown quotient plus an unknown remainder. Specifically, you set up the division equation $45 = 10 \cdot q + r$.

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 $0 \le r < 10$.