# MULTIPLICATION TABLES - REMEMBERING MADE EASY BY SPECIAL TIPS, EASY TO FOLLOW

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Please study the introductory part of
Multiplication
where we discussed the multiplication as repeated addition
with examples and exercises.

There, we also discussed the need to remember
the values of the products of one single digit number
with another single digit number.

Learning these products is much like vocabulary in a language.
Till you master the vocabulary,
the language looks like a foreign language.
Once you master, the language becomes your mother tongue.

Our aim here is to make the single-digits' products
as easy as your mother tongue.

We try to establish the connections between
any two single-digit factors and the resulting product,
until the connection becomes intuitive.

We also provided another page to
help to remember the Multiplication
Table up to 9 x 9 using a simple
principle from Vedic mathematics.
For that page go to
Remembering Multiplication Table
using the principle of Vedic math.

## Multiplication Table upto 10 times10

 x 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

A multiplication table ("times table") is a grid where
rows and columns are headed by the numbers to multiply,
and the entry in each cell is the product
of the column and row headings.
The heading for the first row and first column contains
the symbol "x" which is multiplication operator.

So, for example, 6 x 8 = 48 by looking up where 6 and 8 intersect.

The traditional rote learning of multiplication was based
on memorization of columns in the table, in a form like

```
1 x 8 =  8
2 x 8 = 16
3 x 8 = 24
4 x 8 = 32
5 x 8 = 40
6 x 8 = 48
7 x 8 = 56
8 x 8 = 64
9 x 8 = 72
10 x 8 = 80

```

Learning the contents of the (10x10) table is much easier than it superficially seems to be.

Here are some TIPS which are helpful
in remebering the contents of the Table.

### Tip 1 of Multiplication Tables :

Half of the table is a mirror image of the other
or Order of Multiplying Does Not Matter.

The entries in the Multiplication Table above and below
the Principal Diagonal Elements are the same.

Because of the symmetry of the Multiplication Table
45 entries are in fact duplicates.
(45 + 10 principal diagonal elemements = 55 entries left).

For Example, If you remember 4 x 7 = 28,
then, you also know the value of 7 x 4.
4 x 7 = 28 = 7 x 4

If you remember 3 x 8 = 24,
then, you also know the value of 8 x 3.
3 x 8 = 24 = 8 x 3

Because of Tip1, it is enough if you remember
half of the Table which is shown below.

 x 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 9 12 15 18 21 24 27 30 4 16 20 24 28 32 36 40 5 25 30 35 40 45 50 6 36 42 48 54 60 7 49 56 63 70 8 64 72 80 9 81 90 10 100

### Tip 2 of Multiplication Tables :

1 times table and 10 times table are easy to remember.

1 times any number = that number.
10 times any number = that number followed by zero.

For Example,
1 x 5 = 5; 1 x 7 = 7 etc.
10 x 5 = 50; 10 x 7 = 70 etc.

So, no effort is required in remembering
1 times table and 10 times table.

This reduces the entries of the Table
to remember to the following.

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

### Tip 3 of Multiplication Tables :

We know, the number multiplied by itself
is called the Square of the Number.

The product of the Numbers which differ by 2 is equal to
the square of the in between number minus one.

To apply this Tip, you need to remember
the Squares of the Numbers.

I strongly advise you to remember the
squares of numbers which are given below.

```           1  x  1  =   1
2  x  2  =   4
3  x  3  =   9
4  x  4  =  16
5  x  5  =  25
6  x  6  =  36
7  x  7  =  49
8  x  8  =  64
9  x  9  =  81
10  x 10  = 100
```

Note that these are nothing but
the elements of the principal diagonal.

Now, to apply the above Tip,
consider 7 x 9.
7 and 9 differ by 2.
So their product = 7 x 9
= square of in between number (which is 8) minus 1.
= 8 x 8 - 1 = 64 - 1 = 63.

Let us see another example.
consider 4 x 6.
4 and 6 differ by 2.
So their product = 4 x 6
= square of in between number (which is 5) minus 1.
= 5 x 5 - 1 = 25 - 1 = 24.

With this Tip,
the elements 2 x 4, 3 x 5, 4 x 6, 5 x 7, 6 x 8, 7 x 9
can be remembered and hence the table reduces to

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

### Tip 4 of Multiplication Tables :

The product of the Numbers which differ by 4 is equal to
the square of the middle number minus 4.

Now, to apply the above Tip,
consider 4 x 8.
4 and 8 differ by 4.
So their product = 4 x 8
= square of the middle number (which is 6) minus 4.
= 6 x 6 - 4 = 36 - 4 = 32.

Let us see another example.
consider 3 x 7.
3 and 7 differ by 4.
So their product = 3 x 7
= square of the middle number (which is 5) minus 4.
= 5 x 5 - 4 = 25 - 4 = 21.

With this Tip,
the elements 2 x 6, 3 x 7, 4 x 8, 5 x 9
can be remembered and hence the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 4 6 10 14 16 18 3 9 12 18 24 27 4 16 20 28 36 5 25 30 40 6 36 42 54 7 49 56 8 64 72 9 81 10

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

and remember large chunks of information
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
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playing video games or going out with friends!

## Speed Study System.

### Tip 5 of Multiplication Tables :

Remembering Five times table :

5 times even number = half of the even number followed by 0.
5 times odd number = half of the (odd number - 1) followed by 5.

For example,
5 x 6 = half of 6 followed by 0 = 30
5 x 8 = half of 8 followed by 0 = 40

5 x 7 = half of (7 - 1) followed by 5 = half of 6 followed by 5 = 35
5 x 9 = half of (9 - 1) followed by 5 = half of 8 followed by 5 = 45

With this Tip, the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 4 6 14 16 18 3 9 12 18 24 27 4 16 28 36 5 6 36 42 54 7 49 56 8 64 72 9 81 10

### Tip 6 of Multiplication Tables :

Remembering Nine Times Table made easy :

To remember 9 times table,
which is explained below.

Put your hands in front of you with your palms facing you.
Your fingers represent the numbers one to ten in an order.

1 = left thumb
2 = left index finger
3 = left middle finger
4 = left ring finger
5 = left little finger
6 = right little finger
7 = right ring finger
8 = right middle finger
9 = right index finger
10 = right thumb

To find 4 x 9 :
Close your fourth finger (left ring finger).
The number of fingers to the left of the closed finger (3)
is the left digit
and to the right of the closed finger (6)
is the right digit.

To find 6 x 9 :
Close your sixth finger (right little finger).
The number of fingers to the left of the closed finger (5)
is the left digit
and to the right of the closed finger (4)
is the right digit.

To find 7 x 9 :
Close your seventh finger (right ring finger).
The number of fingers to the left of the closed finger (6)
is the left digit
and to the right of the closed finger (3)
is the right digit.

To find 9 x 9 :
Close your ninth finger (right index finger).
The number of fingers to the left of the closed finger (8)
is the left digit
and to the right of the closed finger (1)
is the right digit.

With this method you can find any value from 1 x 9 to 9 x 9.

Another method to write 9 times table as a whole is,
For the values from 1 x 9 to 10 x 9,
increase the left digit from 0 to 9 and decrease the right digit from 9 to 0.

This is clear from the following vertical presentation.

```           1 x 9 = 09
2 x 9 = 18
3 x 9 = 27
4 x 9 = 36
5 x 9 = 45
6 x 9 = 54
7 x 9 = 63
8 x 9 = 72
9 x 9 = 81
10 x 9 = 90
```

With this Tip, the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 4 6 14 16 3 9 12 18 24 4 16 28 5 6 36 42 7 49 56 8 64 9 10

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### Tip 7 of Multiplication Tables :

onetwo equals three times four i. e. 12 = 3 x 4.
fivesix equals seven times eight i. e. 56 = 7 x 8.

See the order of the numbers 1, 2, 3, 4, 5, 6, 7, 8.

With this Tip, we can remember 3 x 4 and 7 x 8and hence the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 4 6 14 16 3 9 18 24 4 16 28 5 6 36 42 7 49 8 64 9 10

### Tip 8 of Multiplication Tables :

Remembering Two times table :

2 times any number = Adding the number to itself.

Adding the single digit number to itself can be done
with the help of fingers, as explained in Addition.

Here the addition is only once and hence can be done easily.

You may use this tip till you are comfortable in
remembering the 2 times table.

With this Tip, the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 3 9 18 24 4 16 28 5 6 36 42 7 49 8 64 9 10

With these 8 Tips, the 100 elements in the Table are reduced to a mere 9 elements.

### Tip 9 of Multiplication Tables :

Remembering Six times table :

6 times any number = 5 x the number + the number.

Since, we have already seen the Tip (Tip 5) to remember
the 5 times table, we may use that to find 5 x the number
and add the number to get 6 times the number.

For example,
6 x 6 = 6 x 5 + 6 = half of 6 followed by 0 + 6 = 30 + 6 = 36
6 x 8 = 8 x 5 + 8 = half of 8 followed by 0 + 8 = 40 + 8 = 48

6 x 7 = 7 x 5 + 7 = half of (7 - 1) followed by 5 + 7 = half of 6 followed by 5 + 7 = 35 + 7 = 42
6 x 9 = 9 x 5 + 9 = half of (9 - 1) followed by 5 + 9= half of 8 followed by 5 + 9 = 45 + 9 = 54

With this Tip, the table reduces to

 x 1 2 3 4 5 6 7 8 9 10 1 2 3 9 24 4 16 28 5 6 7 49 8 64 9 10

With these 9 Tips, the 100 elements in the Table are reduced to a mere 6 elements.

### Tip 10 of Multiplication Tables :

Applying the Divisibility Rules to the result.

For example, for a number to be divisible by 3
the sum of the digits is to be divisible by 3.

Suppose you write 3 x 8 = 24.
You may check that 2 + 4 = 6 is divisible by 3.

For a number to be divisible by 7,
the difference of two times the last digit and
the remaining number is to be divisible by 7.

Suppose you write 4 x 7 = 28.
You may check that 2 x 8 - 2 = 16 - 2 = 14 is divisible by 7.

Note that all these Tips are only to help
to remember the MultiplicationTables.
If you can directly remember the Multiplication Tables,
these tips are not necessary.

With Tips or without Tips,
you have to remember the Multiplication Tables
upto 9 x 9, accurately.

Put the required time and energy to remember
these Multiplication Tables (1 x 1 to 9 x 9),
without which you can not proceed further
in calculations and in math.

With the help of the above Multiplication table,
you can find the product of any two numbers.

If you still have difficulty,
in remembering the Tables
(1 x 1 to 9 x 9),
Try the Video game mentioned

Some students, who have good memory and
good capacity for attention may remember
the Multiplication Tables upto 20 x 20.

If you can also do the same, here is the
Multiplication Table upto 20 times20.

Note that memorising the Multiplication tables
beyond 9 x 9 is optional.

With the knowledge of Multiplication Tables upto 9 x 9
and the knowledge of Multiplication,
you can find the other entries in the following table.

For example, we have found the values of
4 x 12, 6 x 20, 2 x 30, 4 x 15. 4 x 11 and 5 x 13
in Examples 4 to 9 of Multiplication. Multiplication Table upto 20 times20.

 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

We will see one more Tip here.

Tip 11 of Multiplication Tables :

Rememebering 11 times Table

11 times single digit = digit twice.
11 times two digit number = sum of the digits between the digits, if the sum of digits is single digit..
If the sum of the digits is double digit, carry over 1 to the left.

For Example,
11 x 6 = 66; 11 xx 8 = 88; 11 x 3 = 33;
11 x 12 = 1 (1 + 2) 2 = 132; 11 x 18 = 1 (1 + 8) 8 = 198;
11 x 19 = 1( + 1) 0 9 = 209;
11 x 89 = 8( + 1) 7 9 = 979;

In 11 x 12, 1 + 2 = 3 is put beteen 1 and 2 to get 132.
∴ 11 x 12 = 132.

In 11 x 19, 1 + 9 =10.
So 1 is carried over to the left and 0 is put between the digits.
The carried over 1 added to the left 1 gives 2.
∴ 11 x 19 = 209.

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