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Mental Math Workbook - techniques to multiply a two digit number by a two digit number

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

  1. Addition metnod
  2. Subtraction method
  3. Factorization method

Addition Method

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

Let us say you want to multiply:

x

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

x

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.

Subtraction method

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:

x

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

Note that these techniques work because of the distributive law on multiplication. The distributive law of multiplication states that (a + b) * c = a * c + b * c

Factorization method

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

x

Testing for divisibility

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.

Divisible by 2

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

x

Divisibility by 3

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:

x

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

Divisibility by 9

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.

Divisibility by 5 or 10

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

Divisibility by 11

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:

Divisibility by any odd number

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

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