What is the remainder when the sum of all the odd multiples of five from 1 to 2007 is divided by ten?

Answer 1

Oh, isn't that a happy little math problem! We take the sum of all the odd multiples of five from 1 to 2007, which is 5 + 15 + 25 + ... + 2005. This sum is 100,040,000. When we divide this by ten, we get a remainder of 0, like a peaceful little stream flowing gently through the forest.

Answer 2

The odd multiples of 5 end with 5.

If there is an even number of odd multiples of 5, their sum ends in 0, meaning their remainder when divided by 10 is 0;

If there is an odd number of odd multiples of 5, their sum ends in 5, meaning their remainder when divided by 10 is 5.

By "odd multiples of 5 from 1 to 2007 " do you mean:

• the multiples 1, 3, 5, ..., 2007 of 5?

1 = 2 x 1 - 1, 3= 2 x 2 - 1, ..., 2007 = 2 x 1004 - 1

So there are 1004 multiples of 5, which is even so their sum has a remainder of 0 when divided by 10.

• the numbers between 1 and 2007 which are odd multiples of 5?

1 ÷ 5 = 0.2 → first multiple of 5 in the range is 1 x 5 → the first odd multiple = 1 x 5

2007 ÷ 5 = 401.4 → last multiple of 5 in the range is 401 x 5 → the last odd multiple = 201 x 5

401 = 2 x 201 - 1 which means there are 201 multiples of 5, which is odd so their sum has a remainder of 5 when divided by 10.

As our number system is based on 10, the remainder when dividing by 10 gives the value of the units (last) digit of the original number.

All even multiples of 5 end have a units digit of zero and do not affect the units digit of the sum when added; so the question could have included the even multiples of 5 (ie "What is the remainder when the multiples of five...") without changing the final answer.