2021/034)How do you prove if a, b, and c are integers such that $a|b$ and $c|b$ and a and c are co-primes , then $ac | b$

 Now  $a|b$  so there exists integer d such that $$b = ad\cdots(1)$$

Further  c and a are co-primes so as ber Bezout's identity

There exists integer x and y such that

$cx + ay = 1$

Multiply both sides by d to get

$cx + ayd = d$

or $cx + by = d$ using (1)

as c divides b so there exists integer p such that

$b= pc$

So $cx + pcy = d$

Or $c(x+py) = d$

So from (1) $b = ad = ac(x+py)$ hence ac is factor of b