## Divisible Consecutive Sums

#### Problem

By adding four consecutive integers it is possible to make different totals. For example, 16 + 17 + 18 + 19 = 70, which is also divisible by 10.

How many of the numbers under 100 that are divisible by 10 can you make by adding four consecutive integers?

#### Solution

Taking four consecutive integers, starting from `n`,

n + (n + 1) + (n + 2) + (n + 3) = 4n + 6 |

Clearly for the total (4`n` + 6) to be divisible by 10, 4`n` must end in a 4. So the units digit of `n` must be 1 or 6,

1 + 2 + 3 + 4 = 10 6 + 7 + 8 + 9 = 30 11 + 12 + 13 + 14 = 50 16 + 17 + 18 + 19 = 70 21 + 22 + 23 + 24 = 90 |

Giving 5 numbers under 100 that are divisible by 10 and can be made from the sum of four consecutive integers.

How many numbers under 1000, that are divisible by 5, can be made from the sum of four consecutive integers?

Problem ID: 33 (Feb 2001) Difficulty: 1 Star