Question

When I divide an integer by 15, the remainder is an integer from 0 to 14. When I

divide an integer by 27, the remainder is an integer from 0 to 26.
For instance, if the integer is 100 then the remainders are 10 and 19, which are
different.
How many integers from 1 to 1000 leave the same remainders after division by 15
and after division by 27?

Answer 1

Answer: 119

Explanation:

We know that we must find the least common multiple of 15 and 27 in order to solve the problem because when we are finding remainders that are the same, there must be some relationship between the integer and the two dividing numbers.

Thus, we have the least common multiple 135 and its multiples which will all have a 0 remainder when divided by 15 and by 27.

We can take each of the numbers (7) and the 15 consecutive numbers after each of them, because of modulo becoming the same after 15. If we take the total of all these numbers, which will have the same remainder after division by 15 and 27, we are left with:

15 * 8 - 1 = 119