Learning Only 36 of the Times Tables

Date: 11/04/2001 at 10:39:55
From: Unknown
Subject: Multiplying trick

Hi,

My mother told me of a trick to learn times tables more easily.

She said that I only needed to know 36 of the times tables because of
the problems that can be reversed to a higher problem. 

Example:

   6 x 3 = 18

   3 x 6 = 18

They can be reversed and give the same answer. 

What are the 36 problems that can be reversed to make up the other 
problems? You only have to learn these 36 problems. What are they?

Thank you.

Date: 11/04/2001 at 11:12:33
From: Doctor Sarah
Subject: Re: Multiplying trick

Hi, and thanks for writing to Dr. Math.

Let's look at a multiplication table:


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


Now let's remove the extras. For example, as you noted, once we've 
done 3 x 6, with the 3 on the left and the 6 on top in the table, we 
won't need 6 x 3. Once we've done 7 x 8, we won't need 8 x 7. And so 
on.

     x | 2   3   4   5   6   7   8   9   
     --+-------------------------------
     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 


You can see that you really only need to learn half the table!  

There's another interesting math fact that you can find in this 
version of the table. When you multiply a number by itself, it's 
called "squaring" it. On paper we write 2 squared like this, with a 
smaller 2 above and to the right of the main 2:

     2
    2

Two squared means 2 x 2 and the answer is 4. On a keyboard we use a 
caret like this to write 2 squared:  2^2.

Now let's look at some other squares:

2^2 = 2 x 2 = 4
3^2 = 3 x 3 = 9
4^2 = 4 x 4 = 16
5^2 = 5 x 5 = ?
6^2 = 6 x 6 = ?

Look at the table and find the numbers 4, 9, 16 ...  do you see them 
on the diagonal going from upper left to lower right?  You've found a 
whole line of square numbers!

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/