Summing Multiples Up To a Certain Number

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There are three multiples of 3 less than or equal to 10 – 3, 6 and 9. We don’t have to list the multiples to find how many there are. Instead we could have divided 10 by 3 and rounded down

which rounds down ro 3.

In general, to find the number of multiples ofless than or equal tofindas a mixed fraction or decimal and round the answer down to the nearest whole number.

To find the sum of the multiples of 3 less than or equal to 100, first find how many multiples of 3 there are:

which rounds down to 33, so there are 33 multiples of 3 less than 100.

These multiples of 3 are, in ascending order, 3, 6, 9, …, 99.

This is an arithmetic sequence with first termand common differenceso we can use the formula for the sum of an arithmetic sequence,giving

In fact in general, to find the sum of the multiples of k less than or equal tofind

whereindicates the result of the divisionis rounded down to the nearest whole number.

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