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The Euclidean algorithm finds the greatest common divisor (GCD) of two integers without needing to factor them. The process involves repeatedly performing long division: divide the larger number by the smaller number, then divide the previous divisor by the remainder, and continue this process.

Conditions: Applies to two integers.; Requires repeated long division.; Stops when the remainder is zero.

The Euclidean algorithm finds the greatest common divisor (GCD) of two integers without needing to factor them. The process involves repeatedly performing long division: divide the larger number by the smaller number, then divide the previous divisor by the remainder, and continue this process.

Conditions: Applies to two integers.; Requires repeated long division.; Stops when the remainder is zero.

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 $0 \le r < 10$.

To compute the greatest common divisor of 1785 and 546, apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with 1785 divided by 546.

Conditions: The inputs are 1785 and 546.; The Euclidean algorithm is used.; The division algorithm is applied at each step.

To compute the greatest common divisor of 1785 and 546, apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with 1785 divided by 546.

Conditions: The inputs are 1785 and 546.; The Euclidean algorithm is used.; The division algorithm is applied at each step.

To begin the Euclidean algorithm for $\gcd(1701, 3768)$, place the larger number (3768) on the left side of the division equation and the smaller number (1701) as the divisor. Write $3768 = 1701 \cdot q + r$.

Conditions: The inputs are positive integers.; The larger number is used first on the left-hand side.; The quotient is an integer and the remainder satisfies $0 \le r < 1701$.

To begin the Euclidean algorithm for $\gcd(1701, 3768)$, place the larger number (3768) on the left side of the division equation and the smaller number (1701) as the divisor. Write $3768 = 1701 \cdot q + r$.

Conditions: The inputs are positive integers.; The larger number is used first on the left-hand side.; The quotient is an integer and the remainder satisfies $0 \le r < 1701$.