Question

According to the Fundamental Theorem of Arithmetic, if $p$ (a prime number) divides $b^2$ and $b$ is positive, then _________.

• A. $b$ divides $p$
• B. $b^2$ divides $p$
• C. $p^2$ divides $b^2$
• D. $p$ divides $b$

Answer 1

Answer: (D)

$p$ divides $b$

Let $b=p_1.p_2.p_3.p_4.....p_n$ where $p_1,p_2,p_3,..$ are prime numbers which are necessarily not distinct,

$\Rightarrow b^2=(p_1.p_2.p_3.p_4.....p_n)(p_1.p_2.p_3.p_4.....p_n)$

It is given that $p$ divides $b^2$.

From Fundamental theorem of Arithmetic, we know that every composite number can be expressed as product of unique prime numbers.

This means $p$ belongs to $p_1,p_2,p_3,..p_n$ and is one of them.

Also, $b=p_1.p_2.p_3.p_4...p_n$, thus $p$ divides $b$.