Question

if p is a prime and p divides a2 then p divides -----------where a is a postivite integer

Answer 1

Solution :-

☆ Let a=p1.p2.p3.p4.p5.....pn where, 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.

Hence, it is proved that if pdivides a2 then it also divides a.

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