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Multiplication Facts

Any two numbers from 1 to 9, with the working shown

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Multiplication Facts: the 45 Products Behind Every Times Table

Eighty-one facts that are really forty-five

Two single digits can be multiplied in 81 ways, but the order of the factors does not change the answer: 8 × 3 and 3 × 8 are both 24. Counting each pair once leaves 45 facts — that is the whole of what the times tables hold, and it is half of what the grid looks like.

8 × 3 = 24  ·  3 × 8 = 24  ·  one fact, two readings

The two readings are not the same sentence, though. 8 × 3 is three lots of eight; 3 × 8 is eight lots of three. They arrive at one number by different routes, which is exactly why swapping the factors is a way to check an answer instead of a way to restate it.

Working one out instead of recalling it

Multiplication is repeated addition, so any fact can be rebuilt from scratch: 7 × 4 is 7 + 7 + 7 + 7 = 28. Two shortcuts cover most of the rest.

9 × 6 = 60 − 6 = 54  ·  4 × 7 = 2 × (2 × 7) = 28

The first is the nine rule: multiply by ten and take the number away once. The second is doubling — the four times table is the two times table doubled, and the eight times table is the four doubled again. Between them, the rows that have to be held in memory come down to a handful.

Where this page stops and another starts

A single-digit product is one step with no carrying. As soon as one of the numbers reaches ten the working changes: the answer is assembled from partial products, digit by digit, and that is long multiplication. A whole row at once — one number against 1 to 12 — is a times table.

All 45 facts

1 × 1 = 11 × 4 = 41 × 5 = 51 × 6 = 61 × 7 = 71 × 8 = 82 × 1 = 22 × 2 = 42 × 3 = 62 × 6 = 122 × 9 = 183 × 1 = 33 × 3 = 93 × 5 = 153 × 6 = 184 × 2 = 84 × 3 = 124 × 4 = 164 × 5 = 204 × 7 = 284 × 9 = 365 × 2 = 105 × 5 = 255 × 6 = 305 × 7 = 355 × 9 = 456 × 4 = 246 × 6 = 366 × 7 = 427 × 2 = 147 × 3 = 217 × 7 = 497 × 8 = 568 × 2 = 168 × 3 = 248 × 4 = 328 × 5 = 408 × 6 = 488 × 8 = 648 × 9 = 729 × 1 = 99 × 3 = 279 × 6 = 549 × 7 = 639 × 9 = 81

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FAQ

What is a multiplication fact?

It is one product of two single digits — 8 × 3 = 24, for example. There are 45 of them once the pairs that only differ by order are counted once, and together they are what the times tables hold.

Does the order of the two numbers matter?

Not to the answer: 8 × 3 and 3 × 8 are both 24, because multiplication is commutative. It matters to how the fact is read — 8 × 3 is three lots of eight, 3 × 8 is eight lots of three — and that is exactly why swapping the factors is a way to check the answer.

How is a multiplication fact worked out without memorising it?

By adding: 8 × 3 is 8 + 8 + 8. Two shortcuts cover most of the table — multiplying by 10 and subtracting the number gives the nines (9 × 6 = 60 − 6 = 54), and doubling gives the fours from the twos.

What is the difference between this and long multiplication?

The size of the numbers. A single-digit product is one step and no carries; long multiplication is for numbers of ten and up, where the answer is built from partial products digit by digit.