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
Â
Â
- 225
Â
Â
- 252S
Â
Â
- 256
Â
Â

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 ofÂ r,Â sÂ andÂ t .Â So, in this way we have to approach the problems.

## The correct answer is: 225

### Step by step solution :

It is given thatÂ

That is, at least one ofÂ $p$Â andÂ $q$Â must haveÂ $,Â$Â andÂ Â in their prime factorizations.

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

Case 1 : p containsÂ Â Â then q hasÂ Â Â with K = (0,1,2,3)

That is, number of ways$=4$.

Case 2 : q containsÂ Â Â then q hasÂ Â Â with K = (0,1,2,3)

That is, number of ways$=4$.

Case 3 : Both p and q containsÂ Â

Then, number of ways$=1$.

Therefore, exponent ofÂ $r$Â may be chosen inÂ $2+2+1=5$Â ways.

Similarly,Â exponent of sÂ may be chosen inÂ $4+4+1=9$Â ways andÂ

Exponent ofÂ $s$Â may be chosen inÂ $2+2+1=5$Â ways

Thus, the total number of ways is:

Â

Hence,Â the number of the ordered pairÂ $(p,Âq)$Â isÂ $225$.

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 .Â 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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