Question

If $p$ is a prime number and $p$ divides ${a^2}$, then $p$ divides $a$, if $a$ is a_____
A. irrational
B. consecutive to $p$
C. both A and B
D. positive integer

Answer 1

Hint:
We will take an example to find the answer to this question. Let the prime number be 5 and let $a$ be 25, then we will have 5 divides square of 25 and also 5 divides 25. Since, it satisfies all the conditions, we can see $a = 25$ is a positive number.

Complete step by step solution:
We are given that $p$ is a prime number and $p$ divides ${a^2}$
We will take an example such that it satisfies all the given conditions.
Let the prime number $p$ be 5 as 5 has only 2 factors, 1 and 5.
And let $a = 25$, then 5 divides squares of 25, which is 625.
$\dfrac{{625}}{5} = 125$
Here, 5 also divides 25, that is $p$ divides $a$.
When $p = 5$ and $a = 25$, then all the given conditions are satisfied.
From our example, we can see that $a = 25$ is a positive integer.
Therefore, we can say if $p$ divides ${a^2}$ and $p$ divides $a$, then $a$ is a positive integer.

Hence, option D is correct.

Note:
A number is a prime number if and only if it has only 2 factors, that is 1 and number itself. If any number gets divided by another number, then the square of the number is also divisible by that number. Also, any consecutive number is not divided by its preceding number except for 2.