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Intersection of Sets

If a and b ar...

Question

If a and b are relatively prime, then any common divisor of ac and b is a divisor of c.

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Solution

a and b are relatively prime
$∴ \exists$ integers x and y such that
$a x + b y = 1$ ..........(1)
Let d be any common divisor of ac and b.
since, $d | a c, \exists$ an integer m such that
$a c = d m$ ....(2)
since, $d | b, \exists$ an integer n such that
$b = d n$ ....(3)
Multiplying both sides of (1) by c
$a c x + b c y = c$ ............(4)
Putting the values of ac and b from (2) and (3) in (4).
$d m x + d n c y = c$
or $d (m x + n c y) = c$
$∴ d | c$

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