Permutations and Combinations

Question

Find sum of integers from 1 to 100 that are divisible by 2 or 5.

Answer

Let = Sum of all the numbers from 1 to 100 which are divisible by 2.

= Sum of all the numbers from 1 to 100 which are divisible by 5.

= Sum of all numbers from 1 to 100 which are divisible by both 2 and 5.

∴ Required Sum = ...(i)

Now,

Here,

a + (n - 1)d = 1800 2 + (n - 1)2 = 100

or (n - 1)2 = 98 or n - 1 = or n = 50

∴

Here, a = 5, d = 5,

a + (n - 1)d = 100 or 5 + (n - 1)5 = 100

n - 1 = 19 n = 20

∴

∴ from (i), we have

Required sum = 2550 + 1050 - 550 = 3050.