Question

Let p be a prime number. If p divides a^2,proove that p divides a,where a is an positive integer

Answer 1

Answer on Question #51698 – Math – Combinatorics | Number Theory

Let $p$ be a prime number. If $p$ divides $a^2$, prove that $p$ divides $a$, where $a$ is a positive integer.

Solution

Let $p$ divide $a^2$. Assume that $p$ doesn't divide $a$.

Since $p$ divides $a^2$, then there exists integer $k$ such that $a^2 = pk$. Hence we obtain $a = \frac{pk}{a}$.

Since $a$ is a positive integer, then $\frac{pk}{a}$ is a positive integer. Since $p$ is a prime number then it has only two divisors: 1 and $p$. Due to our assumption $p$ doesn't divide $a$, therefore $GCD(p, a) = 1$. Hence $\frac{k}{a}$ is a positive integer. Assume that $t = \frac{k}{a}$, then $t$ is a positive integer.

Therefore $a = \frac{pk}{a} = pt$, where $t$ is a positive integer, but this means that $p$ divides $a$. So, we come to a contradiction to our assumption.

Thus, $p$ divides $a$.

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