Question

If a number 'x' is divisible by another number 'y', then 'x' is also divisible by all the prime factors of 'y'.

• A. True
• B. False

Answer 1

The correct option is A

True

Suppose x is divisible by y giving a quotient z.

∴ $\frac{x}{y}$= z

or x = y $\times$ z ............(1)
y can be written as a product of its primes as,
$y=p\times q\times r$, where p, q, r are the prime factors of y ...........(2)

Substituting (2) in (1), we get

$x=p\times q\times r\times z$ .........(3)

Dividing (3) by $p$ we get

$\frac{x}{p}=z\times q\times r$

Dividing (3) by $q$ we get

$\frac{x}{q}=z\times p\times r$

Dividing (3) by $r$ we get

$\frac{x}{r}=z\times p\times q$

Hence, $x$ is divisible by all the prime factors of 'y'.