Question:
If two positive integers $m$ and $n$ are expressible in the form $m=p q^{3}$ and $n=p^{3} q^{2}$, where $p, q$ are prime numbers, then $\operatorname{HCF}(m, n)=$
(a) $p q$
(b) pq2
(c) $p^{3} q^{2}$
(d) $p^{2} q^{2}$
Solution:
Two positive integers are expressed as follows:
$m=p q^{3}$
$n=p^{3} q^{2}$
p and q are prime numbers.
Then, taking the smallest powers of p and q in the values for m and n we get
$\operatorname{HCF}(m, n)=p q^{2}$
Hence the correct choice is (b).
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