Question

Let $p,q,r$ be natural numbers. If $m$ is their LCM and $n$ is their HCF, consider the following
$1)$ $mn=pqr$ if each one of $p,q,r$ is prime
$2)$ $mn=pqr$ if $p,q,r$ are relatively prime in pairs

• A. $1$ only
• B. $2$ only
• C. both $1$ and $2$
• D. neither $1$ nor $2$

Answer 1

The correct option is C both $1$ and $2$

$1)$ $p,q,r$ are prime. Let $p$ be $2,q$ be $3$ and $r$ be $5.$

The HCF of $p,q,r$ is $1=n.$ The LCM of $p,q,r$ is $2\times 3\times 5=30=m.$

Also, HCF $\times$ LCM $=30.$ Hence, true.

$2)$ $p,q,r$ are relatively prime in pairs. Let $p$ be $4,q$ be $9$ and $r$ be $25.$

The HCF of $p,q=$ HCF of $q,r=$ HCF of $p,r$ is $1=n.$The LCM of $p,q,r$ is $2\times 3\times 5=900.$

Also HCF $\times$ LCM $=900=m.$

Hence, true.

So, option $C$ is correct .