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# Number Theory

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Use the Euclidean algorithm to find integers x and y  such that  164x + 39y = 1.  State your answer as a list with x first and y second, separated by a comma.

Nov 6, 2021

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Use the Euclidean algorithm to find integers x and y  such that  164x + 39y = 1.  State your answer as a list with x first and y second, separated by a comma.

$$164x+39y=1\\~\\ 164=4*\boxed{39}+8\\ 39=4*\boxed{8}+7\\ 8=1*\boxed{7}+1\\~\\ \text{Now the extended euclidean algorithm}\\ 1=8-1*\boxed{7}\\ 1=8-1*\left[39-4*\boxed{8}\right]\\ 1=8+1\left[-39+4*\boxed{8}\right]\\ 1=5*\boxed{8}-1*39\\ 1=5*\left[ 164-4*\boxed{39} \right]-1*39\\ 1=5*164-21*39\\ 1=164(5)+39(-21)\\ 1=164(5-39s)+39(-21+164a)\\~\\ x=5-39a\qquad y=-21+164a \qquad \text{Where a is any integer.}$$

You can finish it.

LaTex

164x+39y=1\\~\\
164=4*\boxed{39}+8\\
39=4*\boxed{8}+7\\
8=1*\boxed{7}+1\\~\\
\text{Now the extended euclidean algorithm}\\
1=8-1*\boxed{7}\\
1=8-1*\left[39-4*\boxed{8}\right]\\
1=8+1\left[-39+4*\boxed{8}\right]\\
1=5*\boxed{8}-1*39\\
1=5*\left[  164-4*\boxed{39} \right]-1*39\\
1=5*164-21*39\\
1=164(5)+39(-21)\\
1=164(5-39s)+39(-21+164a)\\~\\