Remainders

Not everything shares evenly. 17 shared among 5 gives 3 each, with 2 left over.

The leftover has a name.

Remainder is what is left over after sharing as evenly as possible.

It must be smaller than the divisor

If the remainder is 5 or more when sharing among 5, you could give another one out.

Check by multiplying

3 × 5 = 15, and 15 + 2 = 17. That confirms it.

What a remainder is

Share 17 between 5 and each gets 3, with 2 left over that cannot be shared further as whole items. That 2 is the remainder, written 17 ÷ 5 = 3 r 2. It is not an error or a failure of the division — it is part of the answer.

A remainder is always smaller than the divisor

If you divide by 5, the remainder must be 0, 1, 2, 3 or 4. A remainder of 5 or more means another whole share could have been given out. This is a reliable check on your work: a too-large remainder always signals an incomplete division.

Checking a division with a remainder

Multiply the answer by the divisor and add the remainder: 3 × 5 + 2 = 17. That should return the number you started with. This check works every time and catches both an arithmetic slip and a mishandled remainder.

A remainder of zero

When nothing is left over, the remainder is 0 and we usually do not write it. That case tells you the divisor is a factor of the number — a link to the factor work in the next chapter, and the basis of every divisibility test.