Identifying Multiples of Numbers

Identifying Multiples of Numbers

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: \(2,4,6,8,10,12.\) She has \(12\) shoes in her closet.

The numbers \(2,4,6,8,10,12\) are called multiples of \(2.\) Multiples of \(2\) can be written as the product of a counting number and \(2.\) The first six multiples of \(2\) are given below.

A multiple of a number is the product of the number and a counting number. So a multiple of \(3\) would be the product of a counting number and \(3.\) Below are the first six multiples of \(3.\)

We can find the multiples of any number by continuing this process. The table below shows the multiples of \(2\) through \(9\) for the first twelve counting numbers.

Recognizing the patterns for multiples of \(2,5,10,\text{and}\phantom{\rule{0.2em}{0ex}}3\) will be helpful to you as you continue in this tutorial.

The figure below shows the counting numbers from \(1\) to \(50.\) Multiples of \(2\) are highlighted. Do you notice a pattern?

Multiples of \(2\) between \(1\) and \(50\)

The last digit of each highlighted number in the figure below is either \(0,2,4,6,\text{or}\phantom{\rule{0.2em}{0ex}}8.\) This is true for the product of \(2\) and any counting number. So, to tell if any number is a multiple of \(2\) look at the last digit. If it is \(0,2,4,6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8,\) then the number is a multiple of \(2.\)

Example

Determine whether each of the following is a multiple of \(2\text{:}\)

1. \(\phantom{\rule{0.2em}{0ex}}489\phantom{\rule{0.2em}{0ex}}\)
2. \(\phantom{\rule{0.2em}{0ex}}3,714\)

Solution

1.
Is 489 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8?	No.
	489 is not a multiple of 2.

2.
Is 3,714 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8?	Yes.
	3,714 is a multiple of 2.

Now let’s look at multiples of \(5.\) The figure below highlights all of the multiples of \(5\) between \(1\) and \(50.\) What do you notice about the multiples of \(5?\)

Multiples of \(5\) between \(1\) and \(50\)

All multiples of \(5\) end with either \(5\) or \(0.\) Just like we identify multiples of \(2\) by looking at the last digit, we can identify multiples of \(5\) by looking at the last digit.

The figure below highlights the multiples of \(10\) between \(1\) and \(50.\) All multiples of \(10\) all end with a zero.

Multiples of \(10\) between \(1\) and \(50\)

The figure below highlights multiples of \(3.\) The pattern for multiples of \(3\) is not as obvious as the patterns for multiples of \(2,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.\)

Multiples of \(3\) between \(1\) and \(50\)

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of \(3\) is based on the sum of the digits. If the sum of the digits of a number is a multiple of \(3,\) then the number itself is a multiple of \(3.\) See the table.

Consider the number \(42.\) The digits are \(4\) and \(2,\) and their sum is \(4+2=6.\) Since \(6\) is a multiple of \(3,\) we know that \(42\) is also a multiple of \(3.\)

Look back at the charts where you highlighted the multiples of \(2,\) of \(5,\) and of \(10.\) Notice that the multiples of \(10\) are the numbers that are multiples of both \(2\) and \(5.\) That is because \(10=2\cdot 5.\) Likewise, since \(6=2\cdot 3,\) the multiples of \(6\) are the numbers that are multiples of both \(2\) and \(3.\)