Divisibility Tests

Hello, and welcome to this video about divisibility tests! Today we’ll learn how to figure out if a number is divisible by 2,3,4,5,6,8,9, and 10.

Before we get started, let’s review a few things. When one number is divisible by another, it means that the quotient is a whole number. For instance, 8 is divisible by 4 because \(8\div4=2\). On the other hand, 8 is not divisible by 3 because \(8\div3=2.\overline{66}\).

Divisibility tests are useful ways to determine if an integer is divisible by a divisor without doing a ton of work. These tests involve examining the digits and looking for patterns.

Let’s look at some divisibility tests for some of the most common divisors.

If a number is divisible by 2, then its last digit is an even number, ending in 0, 2, 4, 6, or 8.

For example, 546 is divisible by 2 because it ends in an even number, 6. 545 is not divisible by 2 because it ends in 5, which is not an even number.

If a number is divisible by 3, then the sum of its digits is divisible by 3.

Let’s look at the number 1,635. First, add the digits together: \(1+6+3+5=15\). Next, divide by 3: \(15\div3=5\). Since the sum of the digits, 15, is divisible by 3, 1,635 is also divisible by 3.

Now let’s consider the number 452. To see if 452 is divisible by 3, find the sum of the digits and then divide by 3. \(4+5+2=11\), and \(11\div3=3.\overline{6}\). That means 452 is not divisible by 3.

If a number is divisible by 4, then its last two digits are divisible by 4.

Let’s consider the number 9,316. First, identify the last two digits, 16. Since \(16\div4=4\), 9,316 is divisible by 4.

Now let’s look at the number 45,118. The last two digits of this integer are 18. Since \(18\div4=4.5\), 45,118 is not divisible by 4.

If a number is divisible by 5, then it must end in a 0 or a 5.

For instance, 640 is divisible by 5 because its last digit is 0. On the other hand, 932 is not divisible by 5 because its last digit is neither 0 nor 5.

A number is divisible by 6 if it passes the divisibility tests for 2 and 3 that we talked about earlier. In other words, the number must be even and the sum of its digits must be divisible by 3.

Let’s look at the number 108. We know it’s an even number because its last digit, 8, is even. From here, add the digits together to see if the sum is divisible by 3. \(1+0+8=9\), and \(9\div3=3\). Therefore, 108 is divisible by 6.

Let’s try one more. Consider the number 446. We know it’s an even number because its last digit, 6, is even. From here, add the digits together to see if the sum is divisible by 3. \(4+4+6=14\), and \(14\div3=4.\overline{6}\). Therefore, 446 is not divisible by 6.

A number is divisible by 8 if its last three digits are divisible by 8.

Let’s consider the number 7,184. Divide the last 3 digits, 184, by 8: \(184\div8=23\), so 7,184 is divisible by 8.

Now let’s look at the number 65,447. Divide the last 3 digits, 447, by 8: \(447\div8=55.875\), so 65,447 is not divisible by 8.

A number is divisible by 9 if the sum of its digits is divisible by 9.

Let’s take a look at the number 3,627. First, add the digits together: \(3+6+2+7=18\). Then, divide 18 by 9. \(18\div9=2\), so 3,627 is divisible by 9.

What about 2,157? Is this number divisible by 9? Let’s start by adding the digits together: \(2+1+5+7=15\), and 15 is not divisible by 9. Therefore, 2,157 is not divisible by 9.

A number is divisible by 10 if the number ends in 0.

For example, 890 is divisible by 10 because it ends in 0. 891 is not divisible by 10 because it does not end in 0.

Review

Now that you know the most common divisibility tests, it’s your turn. Consider the following problems:

1. Is 405 divisible by 6?
2. Is 18,524 divisible by 4?
3. Katy says 105 is divisible by 3, but Jamal says it’s not. Who is correct?

Pause the video here and try these problems yourself. When you’re done, resume the video, and we’ll go over everything together.

1. Is 405 divisible by 6?
   To figure out if 405 is divisible by 6, it must be even and the sum of its digits must be divisible by 3. Since 405 is not an even number, it’s not divisible by 6.

 

2. Is 18,524 divisible by 4?
   If 18,524 is divisible by 4, then its last two digits must be divisible by 4. First, identify the last two digits, 24. Since \(24\div4=6\), 18,524 is divisible by 4.

 

3. Katy says 105 is divisible by 3, but Jamal says it’s not. Who is correct?
   To figure out if 105 is divisible by 3, first add together the digits: \(1+0+5=6\). Then, determine if the sum is divisible by 3. Since \(6\div3=2\), Katy is correct.

Great job!

I hope this video about divisibility tests was helpful. Thanks for watching, and happy studying!

Divisibility Test Practice Questions