Question

if P is a prime number and P divides a square then P divides a is it true

Answer 1

Yes it is true by the fundamental theorem of arithmetic

Answer 2

Let a=p1.p2.p3.p4.p5.....pnwhere, p1, p2, p3, ..., pn are prime numbers which are necessarily not distinct.

⇒a2=(p1.p2.p3.p4.p5.....pn).(p1.p2.p3.p4.p5......pn)

 

It is given that p divides a2. From the Fundamental theorem of Arithmetic, we know that every composite number can be expressed as product of unique prime numbers. This means that p is one of the numbers from (p1.p2.p3.p4.p5......pn).

 

We have a=(p1.p2.p3.p4.p5..pn) and p is one of the numbers from (p1.p2.p3.p4.p5......pn).

It means that p also divides a.