Question

If $r$, $s$, $t$ are prime numbers and $p$, $q$ are the positive integers such that LCM of $p$,$q$ is $r^2{t}^4{s}^2$, then the number of ordered pair $(p,q)$ is

• A. $252$
• B. $254$
• C. $225$
• D. $224$

Answer 1

The correct option is C $225$

We need one more additional assumption that $r,s,$ and $t$ are all distinct.

Consider the exponent of $r$ in $p:$

If it is $0or1$, then the corresponding exponent in $q$ is $2$. (two cases)

If it is $2$ then the corresponding exponent in $q$ is $0,1$ or $2.$ (three cases).

Total of five cases for the exponents of r such that we have the given LCM.

$2\times 5-1=9$ cases for the exponents of $t$

$2\times 3-1=5$ cases for the exponents of $s$.

But the ways these three ordered pairs are formed

are completely independent of each other. So the total number of triplets

of ordered pairs of the form $(r,s),(s,t)(t,r)$

Multiply them together to get $5\times 9\times 5=225$ such ordered pairs $(p,q).$