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 of
less than or equal to
find
as 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 term
and common difference
so 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
where
indicates the result of the divisionis rounded down to the nearest whole number.