Exercise 1484Number theory › Divisibility and gcdHigh schoolAdded 2 Nov 2026

A common divisor divides 43

#divisibility#linear combination#prime number

Statement

Let $a=4k+3$ and $b=5k-7$ . Show that any common divisor of $a$ and $b$ divides $43$ .

Hint

Eliminate $k$ by forming a suitable linear combination.

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Elimination. Multiply $a$ by $5$ and $b$ by $4$ :
$5a=20k+15,\qquad 4b=20k-28.$
Subtracting:
$5a-4b=15-(-28)=43.$

Conclusion. Any common divisor $d$ of $a$ and $b$ divides $5a-4b=43$ . Since $43$ is prime, $d\in\{\pm1,\pm43\}$ .