Let a and b be two positive integers such that

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Solution

Q.
Let a and b be two positive integers such that $a = p^{3} q^{4} a n d 𝑏 = 𝑝^{2} q^{3}$ where p and q are prime numbers. If HCF (a,b) = $p^{m} q^{n}$ and LCM (a,b) $= p^{r} q^{s}, t h e n$
(m+n) (r+s) =

1 15

2 30

3 35

4 72

Ans.
(3)

Given, $a = p^{3} q^{4}, b = p^{2} q^{3} ∴ H C F (a, b) = p^{2} q^{3} a n d L C M (a, b) = p^{3} q^{4}$

Comparing with the HCF and LCM given in the question, we get

$H C F (a, b) = p^{m} q^{n} = p^{2} q^{3} \Rightarrow m = 2, n = 3 a n d L C M (a, b) = p^{r} q^{s} = p^{3} q^{4} \Rightarrow r = 3, s = 4 \Rightarrow (m + n) (r + s) = (2 + 3) (3 + 4) = 5 \times 7 = 35$

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