A multiplication table is the set of all products of numbers from 1 to 10. On this page, it is presented in two forms: a 10 × 10 square (Pythagorean table) and in columns for 2, 3, 4… 9 — both are convenient for printing. Additionally, there are techniques that reduce the number of facts to learn by more than half, a finger trick for the nines, the connection with division, and a plan to learn the multiplication table in a week. You can test yourself on the 'Trainer' tab.
Check yourself: 10 variants of 12 problems, hint via neighboring problem.
Open the trainerWhat is multiplication
Multiplication is a shorthand for adding the same number multiple times. Instead of writing \(4 + 4 + 4\), we write \(4 \times 3\): 'four taken three times'.
The numbers being multiplied are called factors, and the result is the product. In the expression \(4 \times 3 = 12\), the factors are 4 and 3, and the product is 12.
It's most convenient to visualize multiplication as a rectangle of dots or cells: how many rows and how many in each. Such a rectangle immediately shows both addition and multiplication, and why the order of factors doesn't change the answer — you just need to rotate the picture.
Multiplication Table: Pythagorean Table from 1 to 10
The complete multiplication table fits into a 10 × 10 square — this is called the Pythagorean table. How to use it: find the first factor in the left column, the second in the top row, and the answer is at their intersection. For example, row 6 and column 7 meet at the cell with 42.
Look closely at the table, and you'll find patterns. Along the diagonal are squares — numbers multiplied by themselves. The table is symmetric: the cell above the diagonal is equal to the cell below it. And each row is a sequence of counting by the same step: row 3 goes 3, 6, 9, 12…
It's convenient to print the table and hang it above your desk: squares and repeating cells are highlighted in color.
| × | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
Multiplication Table by 2, 3, 4, 5, 6, 7, 8, and 9 in Columns
In school, the table is usually learned in columns: first all multiplication by 2, then by 3, and so on. These are the same numbers as in the Pythagorean table, just each row is written out as examples. Columns for 1 and 10 are not shown here: multiplying by 1 doesn't change the number, and multiplying by 10 adds a zero.
Columns are good for learning the table in parts. But if you only answer in order, it's easy to get used to 'counting up' from the beginning. Therefore, once a column is learned, it should be practiced out of order.
How to Learn the Multiplication Table Quickly: Half is Already Known
The table looks like a hundred problems, but you really need to learn much less.
- Commutative Property. \(3 \times 8 = 8 \times 3\). If you've learned one, you already know the other. Out of 100 cells, 55 remain unique — almost half the table for free.
- Multiplying by 0 and 1. Any number multiplied by 0 is 0. Multiplied by 1 — it remains itself.
- Multiplying by 10. Add a zero to the number: \(10 \times 4 = 40\).
- Multiplying by 2 is doubling, adding the number to itself.
- Multiplying by 5 is half of multiplying by 10. All answers end in 0 or 5.
After these rules, only a few problems remain in the columns from 3 to 9. Each of them can be derived from what you already know — the techniques are below.
Multiplication Table by 9: Finger Trick and Digit Sum
The column for 9 is the most 'tricky'. It has two reliable rules.
Digit Sum Rule. In the answers from \(9 \times 2\) to \(9 \times 10\), the sum of the digits is always 9: 18, 27, 36, 45, 54, 63, 72, 81, 90. And the tens digit is one less than the second factor. Therefore, for \(9 \times 4\): the tens digit is 3, and the units digit is such that together they make 9, which is 6. The answer is 36.
Finger Trick. Place your hands in front of you, numbering your fingers from left to right from 1 to 10. To multiply 9 by a number, bend the finger with that number. The fingers to the left of the bent one represent the tens, and those to the right represent the units. This rule works for factors from 1 to 10.
There's a third method, without fingers: multiply by 10 and subtract the number itself.
Multiplication Table by 6, 7, and 8: Squares and Neighbors
These columns are considered the most difficult, but after all the techniques, only a few new problems remain: 6 × 6, 6 × 7, 6 × 8, 7 × 7, 7 × 8, and 8 × 8.
Squares are easier to memorize because there are few of them and they are on the diagonal of the table: 36, 49, 64. Once you know the squares, you can get neighboring answers in one step: \(7 \times 8\) is \(7 \times 7\) plus another seven: \(49 + 7 = 56\). Similarly, \(6 \times 7 = 36 + 6 = 42\).
The second method is through five: \(6 \times 8\) is \(5 \times 8\) plus another eight: \(40 + 8 = 48\).
For 7 × 8, there's also a numerical hint: \(56 = 7 \times 8\) — the digits 5, 6, 7, 8 are in sequence.
Multiplication and Division Table
Division is the inverse operation of multiplication. Therefore, you don't need to learn a separate 'division table': it's already hidden within the multiplication table.
From three numbers in one multiplication fact, you get four correct equations: two for multiplication and two for division. If you know that \(7 \times 8 = 56\), then you also know \(56 \div 8 = 7\) and \(56 \div 7 = 8\).
How to use the Pythagorean table for division: find the dividend in the row of the divisor — the column number will be the answer.
How to Learn the Multiplication Table in a Week: Daily Plan
The order of learning is important: first the simplest columns and rules that give many answers at once, and the difficult problems at the end, when there are few left. It's better to study a little every day, 10–15 minutes, than for a long time once.
Start each session with a short review of yesterday's material. Answer aloud and out of order, not sequentially. Write down mistakes separately: this is your personal list for tomorrow.
It's convenient to test yourself on the 'Trainer' tab of this page: there you can select individual columns — for example, only 6, 7, and 8 — and practice them specifically.
Examples with Explanation: How to Find the Answer if You Forget
Example 1. Forgot 6 × 9 — How to Recall
Method 1 — Through ten. \(10 \times 6 = 60\), subtract one six: \(60 - 6 = 54\).
Method 2 — Through five. \(5 \times 9 = 45\), add another nine: \(45 + 9 = 54\).
Digit sum check: \(5 + 4 = 9\) — the answer from the nines column looks correct.
Answer: \(6 \times 9 = 54\).
Example 2. 4 × 8 by Doubling
Step 1. Multiplying by 4 means doubling twice.
Step 2. \(8 + 8 = 16\).
Step 3. \(16 + 16 = 32\).
Answer: \(4 \times 8 = 32\). The same number is in the cell \(8 \times 4\).
Example 3. Division Using the Table: 72 ÷ 9
Step 1. Ask the multiplication question: what number do you multiply 9 by to get 72?
Step 2. Recall the nines column: in the number 72, the tens digit is 7, so the second factor is one greater — 8.
Step 3. Check: \(9 \times 8 = 72\).
Answer: \(72 \div 9 = 8\).
Example 4. Word Problem: Boxes of Pencils
Problem. One box contains 6 pencils. How many pencils are in 4 boxes?
Step 1. Taking the same amount multiple times is multiplication: \(6 \times 4\).
Step 2. Double twice: \(6 \to 12 \to 24\).
Answer: 24 pencils.
Common Mistakes in the Multiplication Table
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Confusing 7 × 8 and 6 × 8: saying '7 × 8 = 48'.
48 is 6 × 8. The answer for 7 × 8 is one more eight: 48 + 8 = 56. Hint: in the expression 56 = 7 × 8, the digits 5, 6, 7, 8 are in sequence.
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Mixing up answers for 6 × 7, 6 × 8, and 7 × 8 — the numbers 42, 48, and 56 'float'.
Rely on the squares: 6 × 6 = 36, so 6 × 7 = 36 + 6 = 42. 7 × 7 = 49, so 7 × 8 = 49 + 7 = 56. And 6 × 8 = 40 + 8 = 48 using the fives trick. Each of the three numbers can be found using its own shortcut.
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Multiplying by 0, they write the number itself: 5 × 0 = 5.
If you take zero five times, you get zero. Any number multiplied by 0 is 0. A number remains itself only when multiplied by 1.
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Multiplying by 1, they add one: 8 × 1 = 9.
Multiplying by 1 means taking the number once, so 8 × 1 = 8. Adding one is addition, not multiplication.
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Adding instead of multiplying: 4 × 4 = 8.
4 + 4 = 8, but 4 × 4 is four fours: 4 + 4 + 4 + 4 = 16. The confusion comes from 2 × 2 = 2 + 2 = 4 — this is the only case where addition and multiplication yield the same result.
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Knowing the column only in order and recounting from the beginning each time: '7, 14, 21, 28…'.
This is counting, not knowing the table. Once the column is learned, practice it out of order until the answer comes immediately. The 'Trainer' tab presents problems in this way.
Questions and Answers
How to learn the multiplication table quickly?
Don't learn a hundred problems in a row. First, the commutative property and simple rules (×0, ×1, ×2, ×5, ×10) — after these, less than half the table remains. Then, techniques for ×4 and ×9, then squares and neighbors for ×6, ×7, ×8. Study for 10–15 minutes a day and test yourself out of order — this way the table is learned in about a week.
How many multiplication problems do you actually need to learn?
The Pythagorean table has 100 cells, but due to the commutative property, there are only 55 unique problems. If you exclude multiplication by 0, 1, 2, 5, and 10, which are solved by rules, and the nines column with the finger trick, about 15 problems remain. And the most difficult ones, which require a technique, are about a dozen, mainly in the columns for 6, 7, and 8.
What is the difference between the Pythagorean table and the multiplication table in columns?
The numbers are the same. The Pythagorean table is a 10 × 10 square where the answer is at the intersection of a row and column; it clearly shows patterns and symmetry. Columns are the same rows written as problems '7 × 3 = 21'; they are convenient for learning in parts.
How to print the multiplication table?
Open this page and select print in your browser (on a computer, Ctrl+P or ⌘+P). Both the Pythagorean table and the column table will be printed; you can choose not to print unnecessary pages by specifying their numbers in the print dialog.
In what grade is the multiplication table taught?
Usually in 2nd-3rd grade: first, they are introduced to multiplication as adding equal numbers, then they learn the table in columns and immediately connect it with division. The exact order depends on the school curriculum.
Why memorize the table when there's a calculator?
Almost all further mathematics relies on the multiplication table: division, fractions, simplification, column multiplication, equations. If you always count up, your attention shifts from the problem itself to the arithmetic.
How to teach a child the multiplication table without tears?
Short sessions every day, not one long one. Start with what they can do and show that they already know many answers. Praise for speed and correct answers, and repeat difficult problems more often. Games for speed and flashcards out of order work well.
How to check if the multiplication table is learned?
The table is learned when the answer to any problem, given out of order, appears immediately without recounting, and you can also quickly answer the corresponding division problems. You can test yourself on the 'Trainer' tab — it shows which columns have errors.
