To prove 2 odd integers are co-prime

Question

Given two odd integers $a$, $b$ satisfying $pa + qb = 8$ where $p$, $q$ are integers. Prove that $a$ and $b$ are co-prime.

Some numerical examples are:

$3(12) + 7(-4) = 8$. $p = 12, q=-4$ (both even)

$3(-9) + 7(5) = 8$. $p=-9, q=5$ (both odd).

My work so far: a and b are two given odd integers. Let $a=2m+1$, $b=2n+1$ where m, n are integers. Since $pa+qb=8$ --- (1) , therefore $p(2m+1) + q(2n+1) = 8$. => $p + q = 2(4 - mp -nq)$ showing $p + q$ is even.

Case 1: both p and q are even. Let $p = 2p_1, q=2q_1$ where $p_1, q_1$ are integers. Using (1) gives $(2p_1)a + (2q_1) b = 8$ => $(p_1)a +(q_1)b = 4$. Repeating two more similar substitutions give $(p_3)a + (q_3)b = 1$ showing $ax+by=1$ has solution in integers for $x$ and $y$. Hence a and b are co-prime.

Case 2: both $p$ and $q$ are odd integers. I need your help here.

When should I use the condition that both a and b are odd integers? — spring2000, Oct 30 '22 at 05:36
This is a proof by contradiction. 2 is the only prime factor of 8. However, 2 cannot be a prime factor of an odd number. — spring2000, Oct 30 '22 at 08:18
By the dupe $,(a,b) = (a,b,8) = (a,\color{#c00}{(b,8)}) = (a,\color{#c00}1) = 1,$ by $,\color{#c00}{b\ \rm odd}$. — Bill Dubuque, Oct 30 '22 at 13:22

score 5 · Answer 1 · answered Oct 30 '22 at 03:00

5

Suppose $a,b > 1$ are odd numbers which are not coprime. Then they have a common prime factor $p$. Can $p$ divide $8$?

answered Oct 30 '22 at 03:00

catherine

584

To prove 2 odd integers are co-prime

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