Maths-

General

Easy

Question

# If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is , then the number of ordered pair (p, q) is-

- 224
- 225
- 252S
- 256

Hint:

### At least one of the p and q, must have r,s and t in their prime factorization.

Now, we have to consider each case for selecting the number of ways for power of r, s and t . So, in this way we have to approach the problems.

## The correct answer is: 225

### Step by step solution :

It is given that

That is, at least one of $p$ and $q$ must have $,$ and in their prime factorizations.

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

Case 1 : p contains then q has with K = (0,1,2,3)

That is, number of ways$=4$.

Case 2 : q contains then q has with K = (0,1,2,3)

That is, number of ways$=4$.

Case 3 : Both p and q contains

Then, number of ways$=1$.

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

Similarly, exponent of s 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:

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

Be careful about selecting the r s and t for this particular problem. Because number ways for each selection of r, s and t there are 3 cases. Because there are two values given

p, q . Hence to calculate the total number of ways is to multiply the number of ways for r, number of ways for s and number of ways for t. So, in this way to solve this particular type of problem.

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