If are prime numbers and are the positive integers such that LCM of is , then the number of ordered pairs
Consider each case for selecting the power of :
Given that , where are the prime number
At least one of the and should have in prime factorization.
Step 1:Consider the power of
Case 1: If contains , then has with , which implies the number of ways should be
Case 2: If contains then has with , which implies the number of ways should be
Case 3: If both and contains ,which implies the number of ways should be
.Therefore, the power of can be selected in .
Step 2: Consider the power of
Case 1: If contains , then has with , which implies the number of ways should be
Case 2: If contains then has with , which implies the number of ways should be
Case 3: If both and contains ,which implies the number of ways should be
.Therefore, the power of can be selected in .
Step 3: Consider the power of
Case 1: If contains , then has with , which implies the number of ways should be
Case 2: If contains then has with , which implies the number of ways should be
Case 3: If both and contains ,which implies the number of ways should be
.Therefore, the power of can be selected in .
Therefore, the total number of ordered pairs for selecting is given by .
Hence, the correct option is (C).