Question

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

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

Answer 1

The correct option is C

$225$

Consider each case for selecting the power of $r,s,\text{and}t$:

Given that $LCM\left(p,q\right)=r^2{s}^4{t}^2$, where $r,s,t$ are the prime number

At least one of the $p$and $q$ should have $r^2,s^4\text{and}{t}^2$ in prime factorization.

Step 1:Consider the power of $r$

Case 1: If $p$ contains $r^2$, then $q$ has $r^k$ with $k=0,1$, which implies the number of ways should be $$=2$$

Case 2: If $q$ contains $r^2$ then $p$ has $r^k$ with $k=0,1$, which implies the number of ways should be $$=2$$

Case 3: If both $p$ and $q$ contains $r^2$,which implies the number of ways should be $$=1$$

.Therefore, the power of $r$ can be selected in $$2+2+1=5$$.

Step 2: Consider the power of $s$

Case 1: If $p$ contains $$s^4$$, then $q$ has $s^k$ with $k=0,1,2,3$, which implies the number of ways should be $=4$

Case 2: If $q$ contains $$s^4$$ then $p$ has $s^k$ with $k=0,1,2,3$, which implies the number of ways should be $=4$

Case 3: If both $p$ and $q$ contains $$s^4$$,which implies the number of ways should be $$=1$$

.Therefore, the power of $s$ can be selected in $$4+4+1=9$$.

Step 3: Consider the power of $t$

Case 1: If $p$ contains $t^2$, then $q$ has $t^k$ with $k=0,1$, which implies the number of ways should be $$=2$$

Case 2: If $q$ contains $t^2$ then $p$ has $t^k$ with $k=0,1$, which implies the number of ways should be $$=2$$

Case 3: If both $p$ and $q$ contains $t^2$,which implies the number of ways should be $$=1$$

.Therefore, the power of $t$ can be selected in $$2+2+1=5$$.

Therefore, the total number of ordered pairs for selecting $\left(p,q\right)$ is given by $$5\times 9\times 5=225$$.

Hence, the correct option is (C).