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Multiplication table
Multiplication tables are tables showing the result of multiplication on two operands, where the first operand is the multiplier and the second operand is the multiplicand.
The multiplication table for base gives the products where each operand is taken from the digits used in that base, i.e. 0 to . This gives 100 products for base 10, or 64 products if we ignore the trivial cases where either operand is 0 or 1.
0 × 1 = 0 0 × 2 = 0 0 × 3 = 0 0 × 4 = 0 etc. 1 × 1 = 1 1 × 2 = 2 1 × 3 = 3 1 × 4 = 4 etc. 2 × 1 = 2 2 × 2 = 4 2 × 3 = 6 2 × 4 = 8 etc. 3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 3 × 4 = 12 etc. 4 × 1 = 4 4 × 2 = 8 4 × 3 = 12 etc. 5 × 1 = 5 5 × 2 = 10 5 × 3 = 15 etc. 6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 etc. 7 × 1 = 7 7 × 2 = 14 7 × 3 = 21 etc. 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 etc. 9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 etc.
The table can be made much more compact by omitting the operands.
Contents
Multiplication table
Multiplication table for base 10
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The table/matrix diagonal elements are squares (A000290(n), 0 ≤ n ≤ 9). Given that the table/matrix is symmetric about the diagonal, since integer multiplication is commutative, one just needs the lower triangle including the diagonal.
Multiplication table (mod 10) for numbers coprime to 10
× | 1 | 3 | 7 | 9 |
1 | 1 | |||
3 | 3 | 9 | ||
7 | 7 | 1 | 9 | |
9 | 9 | 7 | 3 | 1 |
Reverse multiplication table (mod 10) for numbers coprime to 10
There are t4 = 10 entries in preceding multiplication triangle, so that there are 5 / 2 = 2.5 products for each of the 4 congruences on average.
1 | 1 × 1 | 7 × 3 | 9 × 9 |
3 | 3 × 1 | 9 × 7 | |
7 | 7 × 1 | 9 × 3 | |
9 | 3 × 3 | 7 × 7 | 9 × 1 |
The divisors of an integer which is congruent to {1, 3, 7, 9} modulo 10 (i.e. coprime to 10) are thus of the form:
* 10k + 1 = (10a + 1) (10b + 1) = ab * 10^2 + (1a + 1b + 0) * 10^1 + 1, or (10a + 7) (10b + 3) = ab * 10^2 + (3a + 7b + 2) * 10^1 + 1, or (10a + 9) (10b + 9) = ab * 10^2 + (9a + 9b + 8) * 10^1 + 1; * 10k + 3 = (10a + 3) (10b + 1) = ab * 10^2 + (1a + 3b + 0) * 10^1 + 3, or (10a + 9) (10b + 7) = ab * 10^2 + (7a + 9b + 6) * 10^1 + 3; * 10k + 7 = (10a + 7) (10b + 1) = ab * 10^2 + (1a + 7b + 0) * 10^1 + 7, or (10a + 9) (10b + 3) = ab * 10^2 + (3a + 9b + 2) * 10^1 + 7; * 10k + 9 = (10a + 3) (10b + 3) = ab * 10^2 + (3a + 3b + 0) * 10^1 + 9, or (10a + 7) (10b + 7) = ab * 10^2 + (7a + 7b + 4) * 10^1 + 9, or (10a + 9) (10b + 1) = ab * 10^2 + (1a + 9b + 0) * 10^1 + 9.
Multiplication table (mod 100) for numbers coprime to 100
× | 01 | 03 | 07 | 09 | 11 | 13 | 17 | 19 | 21 | 23 | 27 | 29 | 31 | 33 | 37 | 39 | 41 | 43 | 47 | 49 | 51 | 53 | 57 | 59 | 61 | 63 | 67 | 69 | 71 | 73 | 77 | 79 | 81 | 83 | 87 | 89 | 91 | 93 | 97 | 99 |
01 | 1 | |||||||||||||||||||||||||||||||||||||||
03 | 3 | 9 | ||||||||||||||||||||||||||||||||||||||
07 | 7 | 21 | 49 | |||||||||||||||||||||||||||||||||||||
09 | 9 | 27 | 63 | 81 | ||||||||||||||||||||||||||||||||||||
11 | 11 | 33 | 77 | 99 | 21 | |||||||||||||||||||||||||||||||||||
13 | 13 | 39 | 91 | 17 | 43 | 69 | ||||||||||||||||||||||||||||||||||
17 | 17 | 51 | 19 | 53 | 87 | 21 | 89 | |||||||||||||||||||||||||||||||||
19 | 19 | 57 | 33 | 71 | 9 | 47 | 23 | 61 | ||||||||||||||||||||||||||||||||
21 | 21 | 63 | 47 | 89 | 31 | 73 | 57 | 99 | 41 | |||||||||||||||||||||||||||||||
23 | 23 | 69 | 61 | 7 | 53 | 99 | 91 | 37 | 83 | 29 | ||||||||||||||||||||||||||||||
27 | 27 | 81 | 89 | 43 | 97 | 51 | 59 | 13 | 67 | 21 | 29 | |||||||||||||||||||||||||||||
29 | 29 | 87 | 3 | 61 | 19 | 77 | 93 | 51 | 9 | 67 | 83 | 41 | ||||||||||||||||||||||||||||
31 | 31 | 93 | 17 | 79 | 41 | 3 | 27 | 89 | 51 | 13 | 37 | 99 | 61 | |||||||||||||||||||||||||||
33 | 33 | 99 | 31 | 97 | 63 | 29 | 61 | 27 | 93 | 59 | 91 | 57 | 23 | 89 | ||||||||||||||||||||||||||
37 | 37 | 11 | 59 | 33 | 7 | 81 | 29 | 3 | 77 | 51 | 99 | 73 | 47 | 21 | 69 | |||||||||||||||||||||||||
39 | 39 | 17 | 73 | 51 | 29 | 7 | 63 | 41 | 19 | 97 | 53 | 31 | 9 | 87 | 43 | 21 | ||||||||||||||||||||||||
41 | 41 | 23 | 87 | 69 | 51 | 33 | 97 | 79 | 61 | 43 | 7 | 89 | 71 | 53 | 17 | 99 | 81 | |||||||||||||||||||||||
43 | 43 | 29 | 1 | 87 | 73 | 59 | 31 | 17 | 3 | 89 | 61 | 47 | 33 | 19 | 91 | 77 | 63 | 49 | ||||||||||||||||||||||
47 | 47 | 41 | 29 | 23 | 17 | 11 | 99 | 93 | 87 | 81 | 69 | 63 | 57 | 51 | 39 | 33 | 27 | 21 | 9 | |||||||||||||||||||||
49 | 49 | 47 | 43 | 41 | 39 | 37 | 33 | 31 | 29 | 27 | 23 | 21 | 19 | 17 | 13 | 11 | 9 | 7 | 3 | 1 | ||||||||||||||||||||
51 | 51 | 53 | 57 | 59 | 61 | 63 | 67 | 69 | 71 | 73 | 77 | 79 | 81 | 83 | 87 | 89 | 91 | 93 | 97 | 99 | 1 | |||||||||||||||||||
53 | 53 | 59 | 71 | 77 | 83 | 89 | 1 | 7 | 13 | 19 | 31 | 37 | 43 | 49 | 61 | 67 | 73 | 79 | 91 | 97 | 3 | 9 | ||||||||||||||||||
57 | 57 | 71 | 99 | 13 | 27 | 41 | 69 | 83 | 97 | 11 | 39 | 53 | 67 | 81 | 9 | 23 | 37 | 51 | 79 | 93 | 7 | 21 | 49 | |||||||||||||||||
59 | 59 | 77 | 13 | 31 | 49 | 67 | 3 | 21 | 39 | 57 | 93 | 11 | 29 | 47 | 83 | 1 | 19 | 37 | 73 | 91 | 9 | 27 | 63 | 81 | ||||||||||||||||
61 | 61 | 83 | 27 | 49 | 71 | 93 | 37 | 59 | 81 | 3 | 47 | 69 | 91 | 13 | 57 | 79 | 1 | 23 | 67 | 89 | 11 | 33 | 77 | 99 | 21 | |||||||||||||||
63 | 63 | 89 | 41 | 67 | 93 | 19 | 71 | 97 | 23 | 49 | 1 | 27 | 53 | 79 | 31 | 57 | 83 | 9 | 61 | 87 | 13 | 39 | 91 | 17 | 43 | 69 | ||||||||||||||
67 | 67 | 1 | 69 | 3 | 37 | 71 | 39 | 73 | 7 | 41 | 9 | 43 | 77 | 11 | 79 | 13 | 47 | 81 | 49 | 83 | 17 | 51 | 19 | 53 | 87 | 21 | 89 | |||||||||||||
69 | 69 | 7 | 83 | 21 | 59 | 97 | 73 | 11 | 49 | 87 | 63 | 1 | 39 | 77 | 53 | 91 | 29 | 67 | 43 | 81 | 19 | 57 | 33 | 71 | 9 | 47 | 23 | 61 | ||||||||||||
71 | 71 | 13 | 97 | 39 | 81 | 23 | 7 | 49 | 91 | 33 | 17 | 59 | 1 | 43 | 27 | 69 | 11 | 53 | 37 | 79 | 21 | 63 | 47 | 89 | 31 | 73 | 57 | 99 | 41 | |||||||||||
73 | 73 | 19 | 11 | 57 | 3 | 49 | 41 | 87 | 33 | 79 | 71 | 17 | 63 | 9 | 1 | 47 | 93 | 39 | 31 | 77 | 23 | 69 | 61 | 7 | 53 | 99 | 91 | 37 | 83 | 29 | ||||||||||
77 | 77 | 31 | 39 | 93 | 47 | 1 | 9 | 63 | 17 | 71 | 79 | 33 | 87 | 41 | 49 | 3 | 57 | 11 | 19 | 73 | 27 | 81 | 89 | 43 | 97 | 51 | 59 | 13 | 67 | 21 | 29 | |||||||||
79 | 79 | 37 | 53 | 11 | 69 | 27 | 43 | 1 | 59 | 17 | 33 | 91 | 49 | 7 | 23 | 81 | 39 | 97 | 13 | 71 | 29 | 87 | 3 | 61 | 19 | 77 | 93 | 51 | 9 | 67 | 83 | 41 | ||||||||
81 | 81 | 43 | 67 | 29 | 91 | 53 | 77 | 39 | 1 | 63 | 87 | 49 | 11 | 73 | 97 | 59 | 21 | 83 | 7 | 69 | 31 | 93 | 17 | 79 | 41 | 3 | 27 | 89 | 51 | 13 | 37 | 99 | 61 | |||||||
83 | 83 | 49 | 81 | 47 | 13 | 79 | 11 | 77 | 43 | 9 | 41 | 7 | 73 | 39 | 71 | 37 | 3 | 69 | 1 | 67 | 33 | 99 | 31 | 97 | 63 | 29 | 61 | 27 | 93 | 59 | 91 | 57 | 23 | 89 | ||||||
87 | 87 | 61 | 9 | 83 | 57 | 31 | 79 | 53 | 27 | 1 | 49 | 23 | 97 | 71 | 19 | 93 | 67 | 41 | 89 | 63 | 37 | 11 | 59 | 33 | 7 | 81 | 29 | 3 | 77 | 51 | 99 | 73 | 47 | 21 | 69 | |||||
89 | 89 | 67 | 23 | 1 | 79 | 57 | 13 | 91 | 69 | 47 | 3 | 81 | 59 | 37 | 93 | 71 | 49 | 27 | 83 | 61 | 39 | 17 | 73 | 51 | 29 | 7 | 63 | 41 | 19 | 97 | 53 | 31 | 9 | 87 | 43 | 21 | ||||
91 | 91 | 73 | 37 | 19 | 1 | 83 | 47 | 29 | 11 | 93 | 57 | 39 | 21 | 3 | 67 | 49 | 31 | 13 | 77 | 59 | 41 | 23 | 87 | 69 | 51 | 33 | 97 | 79 | 61 | 43 | 7 | 89 | 71 | 53 | 17 | 99 | 81 | |||
93 | 93 | 79 | 51 | 37 | 23 | 9 | 81 | 67 | 53 | 39 | 11 | 97 | 83 | 69 | 41 | 27 | 13 | 99 | 71 | 57 | 43 | 29 | 1 | 87 | 73 | 59 | 31 | 17 | 3 | 89 | 61 | 47 | 33 | 19 | 91 | 77 | 63 | 49 | ||
97 | 97 | 91 | 79 | 73 | 67 | 61 | 49 | 43 | 37 | 31 | 19 | 13 | 7 | 1 | 89 | 83 | 77 | 71 | 59 | 53 | 47 | 41 | 29 | 23 | 17 | 11 | 99 | 93 | 87 | 81 | 69 | 63 | 57 | 51 | 39 | 33 | 27 | 21 | 9 | |
99 | 99 | 97 | 93 | 91 | 89 | 87 | 83 | 81 | 79 | 77 | 73 | 71 | 69 | 67 | 63 | 61 | 59 | 57 | 53 | 51 | 49 | 47 | 43 | 41 | 39 | 37 | 33 | 31 | 29 | 27 | 23 | 21 | 19 | 17 | 13 | 11 | 9 | 7 | 3 | 1 |
× | 01 | 03 | 07 | 09 | 11 | 13 | 17 | 19 | 21 | 23 | 27 | 29 | 31 | 33 | 37 | 39 | 41 | 43 | 47 | 49 | 51 | 53 | 57 | 59 | 61 | 63 | 67 | 69 | 71 | 73 | 77 | 79 | 81 | 83 | 87 | 89 | 91 | 93 | 97 | 99 |
Reverse multiplication table (mod 100) for numbers coprime to 100
There are t40 = 820 entries in preceding multiplication triangle (mod 100), so that there are 41 / 2 = 20.5 products for each of the 40 congruences on average. (Note the exponential explosion: there are t4 × 10n - 1 entries in the multiplication triangle (mod 10n), so that there are 2 × 10n - 1 + 0.5 products for each of the 4 × 10n - 1 congruences on average.)
01 | 01 × 01 | 43 × 07 | 49 × 49 | 51 × 51 | 53 × 17 | 59 × 39 | 61 × 41 | 63 × 27 | 67 × 03 | 69 × 29 | 71 × 31 | 73 × 37 | 77 × 13 | 79 × 19 | 81 × 21 | 83 × 47 | 87 × 23 | 89 × 09 | 91 × 11 | 93 × 57 | 97 × 33 | 99 × 99 |
03 | 03 × 01 | |||||||||||||||||||||
07 | 07 × 01 | |||||||||||||||||||||
09 | 09 × 01 | |||||||||||||||||||||
11 | 11 × 01 | |||||||||||||||||||||
13 | 13 × 01 | |||||||||||||||||||||
17 | 17 × 01 | |||||||||||||||||||||
19 | 19 × 01 | |||||||||||||||||||||
21 | 21 × 01 | |||||||||||||||||||||
23 | 23 × 01 | |||||||||||||||||||||
27 | 27 × 01 | |||||||||||||||||||||
29 | 29 × 01 | |||||||||||||||||||||
31 | 31 × 01 | |||||||||||||||||||||
33 | 33 × 01 | |||||||||||||||||||||
37 | 37 × 01 | |||||||||||||||||||||
39 | 39 × 01 | |||||||||||||||||||||
41 | 41 × 01 | |||||||||||||||||||||
43 | 43 × 01 | |||||||||||||||||||||
47 | 47 × 01 | |||||||||||||||||||||
49 | 49 × 01 | |||||||||||||||||||||
51 | 51 × 01 | |||||||||||||||||||||
53 | 53 × 01 | |||||||||||||||||||||
57 | 57 × 01 | |||||||||||||||||||||
59 | 59 × 01 | |||||||||||||||||||||
61 | 61 × 01 | |||||||||||||||||||||
63 | 63 × 01 | |||||||||||||||||||||
67 | 67 × 01 | |||||||||||||||||||||
69 | 69 × 01 | |||||||||||||||||||||
71 | 71 × 01 | |||||||||||||||||||||
73 | 73 × 01 | |||||||||||||||||||||
77 | 77 × 01 | |||||||||||||||||||||
79 | 79 × 01 | |||||||||||||||||||||
81 | 81 × 01 | |||||||||||||||||||||
83 | 83 × 01 | |||||||||||||||||||||
87 | 87 × 01 | |||||||||||||||||||||
89 | 89 × 01 | |||||||||||||||||||||
91 | 91 × 01 | |||||||||||||||||||||
93 | 93 × 01 | |||||||||||||||||||||
97 | 97 × 01 | |||||||||||||||||||||
99 | 99 × 01 |
Multiplication triangle
Multiplication triangle rows and antidiagonals
The th row of the multiplication triangle gives the multiples of i.e. all positive multiples of up to . Then, from , you may follow the antidiagonal downwards to get the remaining multiples (you bounce on the right side of the triangle, so to speak).
A002411: Row sums | |||||||||||||||||||||||||||
1 | 1 | ||||||||||||||||||||||||||
2 | 4 | 6 | |||||||||||||||||||||||||
3 | 6 | 9 | 18 | ||||||||||||||||||||||||
4 | 8 | 12 | 16 | 40 | |||||||||||||||||||||||
5 | 10 | 15 | 20 | 25 | 75 | ||||||||||||||||||||||
6 | 12 | 18 | 24 | 30 | 36 | 126 | |||||||||||||||||||||
7 | 14 | 21 | 28 | 35 | 42 | 49 | 196 | ||||||||||||||||||||
8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 288 | |||||||||||||||||||
9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 405 | ||||||||||||||||||
10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 550 | |||||||||||||||||
11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 100 | 121 | 726 | ||||||||||||||||
12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 | 936 | |||||||||||||||
13 | 26 | 39 | 52 | A008587: Multiples 5k of 5, k ≥ 5. | 1183 | ||||||||||||||||||||||
14 | 28 | 42 | A008586: Multiples 4k of 4, k ≥ 4. | 1470 | |||||||||||||||||||||||
15 | 30 | A008585: Multiples 3k of 3, k ≥ 3. | 1800 | ||||||||||||||||||||||||
16 | A005843: Multiples 2k of 2, k ≥ 2. (even numbers) | 2176 | |||||||||||||||||||||||||
A000027: Multiples of 1k of 1, k ≥ 1. (natural numbers) | 2601 |
A075362 Triangle read by rows with the n-th row containing the first n multiples of n.
- {1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 49, 8, 16, 24, 32, 40, 48, 56, 64, 9, 18, 27, 36, 45, 54, 63, 72, 81, 10, 20, ...}
Sequence | A-number | |
---|---|---|
1 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, ...} | A001477 |
2 | {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, ...} | A005843 |
3 | {0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, ...} | A008585 |
4 | {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, ...} | A008586 |
5 | {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, ...} | A008587 |
6 | {0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, ...} | A008588 |
7 | {0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, ...} | A008589 |
8 | {0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, ...} | A008590 |
9 | {0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, ...} | A008591 |
10 | {0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, ...} | A008592 |
Multiplication triangle row sums
The row sums give the pentagonal pyramidal numbers
where is the th triangular number.
A002411 Pentagonal pyramidal numbers: n^2*(n+1)/2.
- {0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, 11368, 12615, 13950, ...}
Sequences
A004247 Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0).
- {0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, ...}
A027424 Number of distinct products ij with 1 <= i, j <= n (number of distinct terms in n X n multiplication table).
- {1, 3, 6, 9, 14, 18, 25, 30, 36, 42, 53, 59, 72, 80, 89, 97, 114, 123, 142, 152, 164, 176, 199, 209, 225, 239, 254, 267, 296, 308, 339, 354, 372, 390, 410, 423, 460, 480, ...}