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
b2 divides p
C
p2 divides b2
D
p divides b

Solution

## The correct option is C $$p$$ 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 $$b$$.Mathematics

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