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# Multiplication table

(Redirected from Multiplication triangle)

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 ${\displaystyle b}$ gives the ${\displaystyle b^{2}}$ products where each operand is taken from the ${\displaystyle b}$ digits used in that base, i.e. 0 to ${\displaystyle b-1}$. 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.

## Multiplication table

### Multiplication table for base 10

 × 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81

 × 0 1 2 3 4 5 6 7 8 9 0 0 1 0 1 2 0 2 4 3 0 3 6 9 4 0 4 8 2 6 5 0 5 0 5 0 5 6 0 6 2 8 4 0 6 7 0 7 4 1 8 5 2 9 8 0 8 6 4 2 0 8 6 4 9 0 9 8 7 6 5 4 3 2 1

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 ${\displaystyle k}$th row of the multiplication triangle gives the multiples ${\displaystyle ik,\,1\leq i\leq k,}$ of ${\displaystyle k,\,k\geq 1,}$ i.e. all positive multiples of ${\displaystyle k}$ up to ${\displaystyle k^{2}}$. Then, from ${\displaystyle k^{2}}$, you may follow the antidiagonal downwards to get the remaining multiples ${\displaystyle ik,\,i\geq k+1}$ (you bounce on the right side of the triangle, so to speak).

 A002411: Row sums${\displaystyle k\cdot \sum _{i=1}^{k}i=}$${\displaystyle {\frac {k^{2}(k+1)}{2}}}$ 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${\displaystyle (n),\,n\geq 5}$: Multiples 5k of 5, k ≥ 5. 1183 14 28 42 A008586${\displaystyle (n),\,n\geq 4}$: Multiples 4k of 4, k ≥ 4. 1470 15 30 A008585${\displaystyle (n),\,n\geq 3}$: Multiples 3k of 3, k ≥ 3. 1800 16 A005843${\displaystyle (n),\,n\geq 2}$: Multiples 2k of 2, k ≥ 2. (even numbers) 2176 A000027${\displaystyle (n),\,n\geq 1}$: 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, ...}

Multiples ${\displaystyle nk,\,n\geq 0,}$ of ${\displaystyle k}$
${\displaystyle k}$ 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

${\displaystyle k\cdot \sum _{i=1}^{k}i=k\cdot t_{k}=k\cdot {\frac {k(k+1)}{2}}={\frac {k^{2}(k+1)}{2}},}$

where ${\displaystyle t_{k}}$ is the ${\displaystyle k}$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, ...}