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)

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