To solve the problem of finding the number of ways to select 2 distinct positive integers (a) and (b) such that (1 \leq a < b \leq 100) and the product (ab) is divisible by 5, follow these steps:

Step 1: Calculate the total number of pairs

The total number of ways to choose 2 distinct numbers from 1 to 100 is given by the combination formula (C(n,2)):
[C(100,2) = \frac{100 \times 99}{2} = 4950]

Step 2: Calculate the number of pairs where (ab) is NOT divisible by 5

For (ab) not to be divisible by 5, neither (a) nor (b) can be divisible by 5.

• Number of numbers between 1 and 100 divisible by 5: (\frac{100}{5} = 20)
• Number of numbers not divisible by 5: (100 - 20 = 80)

The number of pairs from these 80 numbers is:
[C(80,2) = \frac{80 \times 79}{2} = 3160]

Step 3: Subtract non-divisible pairs from total pairs

The number of valid pairs is:
[4950 - 3160 = 1790]

Answer: (\boxed{1790})

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