Question

If $r,s,t$ are prime numbers and $p,q$ are the positive integers such that the 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 $225$

It is given that LCM$(p,q)=r^2{t}^4{s}^2$.

That is, at least one of $p$ and $q$ must have $r^2,t^4$ and $s^2$ in their prime factorizations.

Now, consider the cases for power of $r$ as follows:

Case 1: $p$ contains $r^2$ then $q$ has $r^k$ with $k=(0,1)$.

That is, number of ways$=2$.

Case 2: $q$ contains $r^2$ then $p$ has $r^k$ with $k=(0,1)$.

That is, number of ways$=2$.

Case 3: Both $p$ and $q$ contains $r^2$

Then, number of ways$=1$.

Therefore, exponent of $r$ may be chosen in $2+2+1=5$ ways.

Similarly, exponent of $t$ may be chosen in $4+4+1=9$ ways and exponent of $s$ may be chosen in $2+2+1=5$ ways

Thus, the total number of ways is:

$5\times 9\times 5=225$

Hence, the number of the ordered pair $(p,q)$ is $225$.