Question

How many integers from 1 to 500 are divisible by 3 or 5?

Answer 1

Your Answer is 266.

Thanks.

Answer 2

Given:

The set of integers from 1 to 500.

To Find:

The number of integers from 1 to 500 such that it is divisible by 3 or 5.

Solution:

The given problem can be solved using divisibility concepts.

1. For a number to be divisible by 3, the sum of the digits of the number must be divisible by 3.

2. For a number to be divisible by 5, the units digit of the number must be either 0 or 5.

3. The Total number of integers from 1 to 500 which are divisible by 3 or 5 = (total number of integers from 1 to 500 divisible by 3) +(total number of integers from 1 to 500 divisible by 5) - (total number of integers from 1 to 500 which are divisible by both 3 and 5).

4. The numbers 3, 6, 9, 12, 15, .., 498 are divisible by 3. ( The series is in an AP )

=> Total number of terms can be calculated using the formula, $T_{n} = a + (n-1)d$, ( where a = first term, d = common difference, Tn = last term )

=> 498 = 3 + 3n - 3,

=> n = 166.

5. The numbers 5, 10, 15, 20, .. , 500 are divisible by 5. ( The series is also in A.P )

=> 500 = 5 + 5(n-1),

=> 500 = 5n,

=> n = 100.

6. The numbers 15, 30, 45, ... 495 are divisible by 3 and 5. ( The series is also in an A.P )

=> 495 = 15 + 15(n-1),

=> 495 = 15 + 15n -15,

=> 15n = 495,

=> n = 33.

7. The Total number of integers from 1 to 500 which are divisible by 3 or 5 = (total number of integers from 1 to 500 divisible by 3) + (total number of integers from 1 to 500 divisible by 5) - (total number of integers from 1 to 500 which are divisible by both 3 and 5).

=> The Total number of integers from 1 to 500 which are divisible by 3 or = 166 + 100 - 33,

=> 266 - 33,

=> 233.

Therefore, the number of integers is 233.