Algebra Solutions: If an integer divides the greatest common divisor of two integers, then it divides any linear combination of the two integers

If an integer divides the greatest common divisor of two integers, then it divides any linear combination of the two integers

Let $a,b,k$ be integers. Prove that if $k$ divides $a$ and $b$ , then $k$ divides $as + bt$ for all integers $s$ and $t$ .

Suppose $k$ divides $a$ and $b$ ; then there exist integers $n$ and $m$ such that $a = kn$ and $b = km$ . Now we have $as + bt = kns + kmt = k(ns + mt)$ for all $s,t$ , so $k$ divides $as + bt$ .

Algebra Solutions

If an integer divides the greatest common divisor of two integers, then it divides any linear combination of the two integers

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