Question

1. Use the Euclidean Algorithm to find integers \( a \) and \( b \) such that \[ 81 a+52 b=1 \]

Answer 1

The integers
`\( a = -9 \) and \( b = 14 \)`satisfy the equation
`\( 81a + 52b = 1 \).`

Step-by-step explanation:

To find integers ( a ) and ( b ) satisfying ( 81a + 52b = 1 ) using the Euclidean Algorithm, we initiate the process by expressing 81 and 52 as a linear combination in terms of each other. Beginning with the equation
`( 81 = 52 * 1 + 29 \),`we rewrite it as ( 81 = 52 \times 1 + 29 \),
`( 81 = 52 * 1 + 29 \),`. Proceeding iteratively, we express 52 as a linear combination of 29 and 81, repeating the process until we attain a form where the greatest common divisor of 81 and 52 equals 1.

Continuing the process, we eventually reach the equation
`\( 1 = 29 - 1 * 2 \) where \( 29 \)`is expressed as a linear combination of
`\( 81 \) and \`( 52 \). Rearranging terms leads us to
`\( 1 = 29 - (81 - 52 * 1) \),`simplifying to
`( 1 = -81 + 52 * 2 \)`. Consequently, by matching coefficients, we find that ( a = -9 ) and ( b = 14 ) satisfy the equation ( 81a + 52b = 1 ), demonstrating the solution derived through the Euclidean Algorithm.

This process showcases how the Euclidean Algorithm systematically computes the coefficients of the linear combination, ultimately solving for
`\( a \) and \( b \)` such that their combination results in the desired value. The obtained values satisfy the equation
`\( 81a + 52b = 1 \)`, confirming the accuracy of the method used to determine the integers
`\( a \) and \( b \).`