Question

If the LCM of $p,q$ is $r^2{t}^4{s}^2$, where $r,s,t$ are prime numbers and $p,q$ are the positive integers then the number of ordered pair $\left(p,q\right)$ is equal to:

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

Answer 1

The correct option is C

$225$

Explanation for the correct option.

Step 1: Find the number of ways of choosing exponents of $r$.

The LCM of $p,q$ is $r^2{t}^4{s}^2$, it means $r^2,t^4,s^2$ must be among the prime factors of $p,q$.

Now, if $r^2$ is a factor of $p$, then $q$ has $r^x$, where $x=\left(0,1\right)$

So, the number of ways of choosing exponents of $r$ will be $2$.

Similarly, if $r^2$ is a factor of $q$, then $p$ has $r^x$, where $x=\left(0,1\right)$

So, the number of ways ways of choosing exponents of $r$ will again be $2$.

But if if $r^2$ is a factor of both $p,q$, then the number of ways ways of choosing exponents of $r$ will be $1$.

So, the exponents of $r$ can be chosen in $2+2+1=5$ ways.

Step 2: Find the number of ways of choosing exponents of $t$.

If $t^4$ is a factor of $p$, then $q$ has $t^x$, where $x=\left(0,1,2,3\right)$

So, the number of ways of choosing exponents of $t$ will be $4$.

Similarly, if $t^4$ is a factor of $q$, then $p$ has $t^x$, where $x=\left(0,1,2,3\right)$

So, the number of ways ways of choosing exponents of $t$ will again be $4$.

But if if $t^4$ is a factor of both $p,q$, then the number of ways ways of choosing exponents of $t$ will be $1$.

So, the exponents of $t$ can be chosen in $4+4+1=9$ ways.

Step 3: Find the number of ways of choosing exponents of $s$.

If $s^2$ is a factor of $p$, then $q$ has $s^x$, where $x=\left(0,1\right)$

So, the number of ways of choosing exponents of $s$ will be $2$.

Similarly, if $s^2$ is a factor of $q$, then $p$ has $s^x$, where $x=\left(0,1\right)$

So, the number of ways ways of choosing exponents of $s$ will again be $2$.

But if if $s^2$ is a factor of both $p,q$, then the number of ways ways of choosing exponents of $s$ will be $1$.

So, the exponents of $s$ can be chosen in $2+2+1=5$ ways.

Step 4: Find the number of ordered pair $\left(p,q\right)$.

The number of ordered pair $\left(p,q\right)$ is $5\times 9\times 5=225$

Hence, option C is correct.