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-

Hint:

## The correct answer is: 225

### Step by step solution :It is given that That is, at least one of  and  must have  and  in their prime factorizations.Now, consider the cases for power of  as follows:Case 1 : p contains   then q has   with K = (0,1,2,3)That is, number of ways.Case 2 : q contains   then q has   with K = (0,1,2,3)That is, number of ways.Case 3 : Both p and q contains  Then, number of ways.Therefore, exponent of  may be chosen in  ways.Similarly, exponent of s may be chosen in  ways and Exponent of  may be chosen in  waysThus, the total number of ways is: Hence, the number of the ordered pair  is .

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.