let p be a prime number If p divides a square,prove p divides a,where a is a positive integer - Maths - Real Numbers

Question

let p be a prime number If p divides a square,prove p divides
a,where a is a positive integer

Ajanta Trivedi · Accepted Answer

to prove that if p is a prime number and if p divides $a^{2}$ , then p divides a.

so first we will prove Euclid' Lemma: i.e.

" if p is a prime and p divides ab, then p divides a or p divides b"

proof: assume that p does not divide a . then we must show that p divides b.

hcf of p and a can be written as linear combination of a and p. there exist two integers s and t such that

$s p + t a = (p, a)$

since p does not divide a. therefore hcf of p and a is 1.

therefore $s p + t a = 1.......... (1)$

multiplying the eq(1) by b.

$s p b + t a b = b$ ...........(2)

since p divides ab therefore p divides tab and

p divides itself , then p divides spb.

thus p divides $s p b + t a b$

then from eq(2): p divides b.

thus if p divides ab then p divides a or p divides b.

now put b = a.

thus if p divides $a^{2} = a * a$ , then p divides a.

hope this helps you.

cheers!!

let p be a prime number.If p divides a square,prove p divides a,where a is a positive integer.