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})
作者声明:本文包含人工智能生成内容。
