Question

Let product of two natural numbers $\alpha, \beta$ is 192 . Their greatest common divisor and least common multiple is represented by p and q. If ratio of A.M. and H.M. of p and q is $\frac{169}{48}$, then $\alpha+\beta$ can be

• A. 52
• B. 28
• C. 98
• D. 32

Answer 1

Answer: (A), (B)

$\therefore p q =192 $
$\because \frac{ A \cdot M .}{ H \cdot M .}=\frac{\frac{ p + q }{2}}{\frac{2 pq }{ p + q }}=\frac{( p + q )^2}{4 pq }=\frac{169}{48} $
$\Rightarrow( p + q )^2=\frac{169 \times 4 \times 192}{48}=13^2 \times 4^2$
$\Rightarrow p + q =52 \Rightarrow q =52- p$
$\therefore p (52- p )=192 \Rightarrow p ^2-52 p +192=0 $
$\Rightarrow p =4 \text { or } 48$
$\therefore q =48$ or 4
$\therefore HCF =4$ and $LCM =48$
$\therefore$ Numbers $\alpha$ and $\beta$ can be 4 and 48 or 12 and 16