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Multiples of 3 and 5



If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

It is easy to find the answer using computer, but it's more interesting if we can solve it without coding.

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The multiples of 3 are all of the form 3n where n is a natural number. The largest value of n is floor(1000/3) or 333. So we get 3(1), 3(2), 3(3), ..., 3(333). So if we sum them...sum[n=1...333](3n) we get 3* (333*334)/2 or 166,833.

The same follows for 5 1000/5=200. So sum[n=1...200](5n) or 5*(200*201)/2 or 100,500.

Finally we add them together to get 267,333.

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