Essentially, there are three methods which we can apply to simplify a two-digit multiplication.

This techinque works when the second digit of the multiplicand is close to zero.

Let us say you want to multiply:

We can break the problem into two simpler multiplications and one addition, like this:

The reason we apply this technique to multipliers where the units digit is small is that the second step is simple to calculate. In the above instance, it is easy to multiply 23 by 2.

If the units digit in the multiplicand is close to 9, we can break the problem into two simpler multiplications and one subtraction, like this:

Splitting the problem as above makes the multiplication simpler because of two reasons:

- We can do the calculations left to right.
- The units digit in the first step is 0, thereby simplifying the multiplication to a single digit multiplication (e.g. 57 is broken down as 50 and 7.)

Note that these techniques work because of the distributive law on multiplication. The distributive law
of multiplication states that

In the factorization method, we break the multiplier into factors, such that the entire multiplication splits up into two single digit multiplications.

In order to master the factorization method, you need to be able to quickly test for divisibility. We will outline these in the following sections.

A number will be divisible by 2 if it ends in 2, 4, 6, 8 or 0. We can apply this as in the following example

To test if a number is divisible by 3, sum up all the digits and if the sum is divisible by 3, then the original number is divisible by three. This rule applies to any number having any number of digits.

For example, to test if 12345 is divisible by 3, we sum up the digits 1,2,3,4,5:

and since the sum, 15, is divisible by 3, the number 12345 is divisible by 3.

To test if a number is divisible by 9, sum up all the digits and if the sum is divisible by 9, then the original number is divisible by three. This rule applies to any number having any number of digits.

Any number that ends in 5 or 0 is divisible by 5. This rule applies to any number having any number of digits.

To test if a number is divisible by 11, alternative subtract and add the digits of the number. If the result is either 0 or if it is a multiple of 11, then the number is divisible by 11. For example:

- To test if the number 6446 is divisible by 11, (6-4)+(4-6) = 0. So, 6446 is divisible by 11.
- To test if the number 7035 is divisible by 11, (7-0)+(3-5) = 5. So, 7035 is not divisible by 11.
- To test if 8,492 is divisible by 11, (8-4)+(9-2) = 11. So, 8,492 is divisible by 11.
- To test if 94162 is divisible by 11, (9-4)+(1-6)+2 = 2, so 94162 is not divisible by 11.

To test if a number is divisible by an odd number, say 7, follow the following steps.

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