Division

Updated 7 October 2026

Division has a reputation, and it is unearned. Every division question is a multiplication question read backwards: 72 ÷ 8 is just "8 times what makes 72?" If you know your times tables, you already hold the answers.

What division means (two pictures)

Sharing: 12 sweets between 3 friends — how many each? Grouping: 12 sweets, bags of 3 — how many bags? Both are 12 ÷ 3 = 4, and both pictures matter: sharing is how children meet division, grouping is the one that scales up (long division is grouping, done systematically).

How do you divide in your head?

To divide in your head, turn the division into a times-table question: 54 ÷ 6 asks what number times 6 makes 54, and 6 × 9 = 54, so the answer is 9. Every times-table fact you know is also a division fact, because division undoes multiplication. For numbers past the tables, split the number into a friendly multiple of the divisor and a part you know. For 78 ÷ 6, take 60, which is ten 6s, and 18 is left, which is three 6s, so 78 ÷ 6 = 10 + 3 = 13. Dividing by 4 or 8 has its own route: halve twice for 4 and three times for 8, so 96 ÷ 8 goes 48, 24, 12. When a number does not share out exactly, the leftover is a remainder, and it must be smaller than the divisor. To check any answer, multiply back: 13 × 6 = 78.

Step 1: read the times table backwards

$56÷7\to$56 ÷ 7 → "seven times what is 56?" $\to 8$→ 8. No new skill; a fact you own, read in the other direction.

Step 2: divide by 4 and 8 by halving

1. ÷ 4 = halve, then halve again. $156÷4\to 78\to 39$156 ÷ 4 → 78 → 39.
2. ÷ 8 = halve three times. $328÷8\to 164\to 82\to 41$328 ÷ 8 → 164 → 82 → 41.

Step 3: divide by splitting

Peel off an easy multiple of the divisor, share the rest.

$91÷7\to$91 ÷ 7 → peel off 70 (ten $7s)\to 21$7s) → 21 left (three $7s)\to 10+3=$7s) → 10 + 3 = 13.

How do you do short division, digit by digit?

Short division works through the number from left to right, one digit at a time: divide the digit by the divisor, write how many whole times it fits above it, and carry any remainder onto the front of the next digit. For 738 ÷ 3, 7 ÷ 3 is 2 with 1 over, so write 2 and carry the 1 to make 13. Next, 13 ÷ 3 is 4 with 1 over, which makes 18. Last, 18 ÷ 3 = 6 exactly, so 738 ÷ 3 = 246. It works because each carried 1 is worth ten of the next place: the leftover hundred becomes 10 tens. When the first digit is smaller than the divisor, start one digit later. In 256 ÷ 4, the 2 is too small, so write nothing above it and divide 25 instead: 6 with 1 over, then 16 ÷ 4 = 4, giving 64. To check, multiply back: 246 × 3 = 738.

852 ÷ 4:

1. 8 ÷ 4 = 2, nothing left over.
2. 5 ÷ 4 = 1, remainder 1 — the 1 carries onto the front of the next digit.
3. That makes the last digit 12. 12 ÷ 4 = 3.

Answer: 213. Check: 213 × 4 = 852. ✓

Long division, without fear

Long division is short division with the leftovers written out instead of carried in your head — which is exactly why it works when the divisor has two digits and your head runs out of room. Four moves, repeated until done:

1. Divide — how many times does the divisor fit? Write it on top.
2. Multiply — how much did that use up?
3. Subtract — what is left?
4. Bring down the next digit, and repeat.

936 ÷ 24:

1. 24 into 93 fits 3 times (3 × 24 = 72). Write 3.
2. 93 − 72 = 21 left.
3. Bring down the $6\to 216$6 → 216.
4. 24 into 216 fits 9 times exactly (9 × 24 = 216). Write 9. Nothing left.

Answer: 39. Check: 39 × 24 = 936. ✓

What is a remainder?

A remainder is the amount left over when a number cannot be shared into equal groups exactly. 17 ÷ 5 = 3 remainder 2 Five fits into 17 three times, which uses 3 × 5 = 15, and 17 − 15 = 2 is left over. A remainder is always smaller than the divisor, because if 5 or more were left, one more group of 5 would fit. So 17 ÷ 5 can never leave 6, and a remainder that big means the answer above it is too small. What you do with the remainder depends on the question. If 17 people travel in cars that hold 5 each, the 2 left over still need a seat, so the answer is 4 cars, not 3. If 17 pencils are packed in boxes of 5, only 3 boxes are full, and 2 pencils stay loose. To check a remainder, multiply and add it back: 3 × 5 + 2 = 17.

Money is different: $17 shared by 5 is $3.40 each (keep dividing into decimals).

How can you tell a number divides without dividing?

A divisibility test reads the digits instead of dividing. A number divides by 2 if its last digit is even, by 5 if it ends in 0 or 5, and by 10 if it ends in 0. For 3 and 9, add the digits: the number divides by 3 when the digit sum does, and by 9 when the digit sum does. Take 4,158. Its last digit is 8, so it divides by 2. Its digits add to 4 + 1 + 5 + 8 = 18, and 18 divides by both 3 and 9, so 4,158 does too. The last-digit tests work because every ten already divides by 2, 5 and 10, so only the ones digit can spoil it. The digit-sum test works because 10, 100 and 1,000 are each one more than a multiple of 9. If the digit sum is still large, add its digits again. To check, divide once: 4,158 ÷ 9 = 462.

Every test, with the reason it works: the divisibility rules.