Question

If $r,s$ and $t$ are prime numbers and $p,q$ are 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. 254
• B. 252
• C. 225
• D. 224

Answer 1

The correct option is C 225

If 2 numbers A and B have LCM as L, where L = $a^x\times b^y\times c^z$ ., then, number of ordered pairs of numbers having the above LCM will be: $(2x+1)\times (2y+1)\times (2z+1)$

Let us consider r: we need the power r in either p or q to be at least $2$.

If the power of r in p is 2, then in q it should be 0 or 1 or 2 $\to 3$ cases

If the power of r in q is 2, then in q it should be 0 or 1 or 2 $\to 3$ cases

But (2, 2) has been counted twice. Thus, there are $3+3-1=5$ cases

Total of five cases for the exponents of r such that we have the given LCM which is $2\times 2+1$

Similarly, the others follow.
Here, $x=2,y=4,z=2.$

Therefore, the answer = $(2\times 2+1)\cdot (2\times 4+1)\cdot (2\times 2+1)=225$

Hence, (C) is correct.