Divisibility Rule of 17: Examples and FAQ – Mindspark

A number is divisible by 17 if it leaves zero as the remainder when divided by 17. Here we will understand the divisibility rule of 17 with some examples.

The Divisibility Rule of 17

How do we know if a number is wholly divisible by 17 or not? Usually, we check this with the lengthy mathematical division process. But to make things easy, the divisibility rule of 17 has a shortcut method to tell if a number is divisible by 17 or not. 

There are 3 rules that we follow – 

Rule 1

Multiply the last digit by 5 and subtract that from the rest of the number. If that result is either a zero or a number divisible by 17, we can confirm that the number is divisible by 17.

Example:   

For 969, we do: 96 – (9 x 5) = 96 – 45 = 51. Since 51(51 = 17 x 3) is divisible by 17.

Hence 969 is also divisible by 17.

Rule 2

Take the number formed by the last two digits of the given number. Multiply the rest of the number by 2. Now subtract the product from the number formed by the last two digits. If that result is either a zero or a number divisible by 17, then the given number is divisible by 17.

Example: 

For 4675, we do: (46 × 2 ) – 75 = 17.

Since the result is 17 itself, 4675 is divisible by 17.

Rule 3

Find the sum of 9 times the last digit to 5 times the rest of the number. If the sum is either a zero or a number divisible by 17, we confirm that the number is divisible by 17.

Example:  

For 986, we do: (98 × 5) + (6 × 9) = 490 + 54 = 544

Now for 544, (54 × 5) + (4 × 9) = 270 + 36 = 306

Now for 306, (30 × 5) + (6 × 9) = 150 + 54 = 204

Now for 204, (20 × 5) + (4 × 9) = 100 + 36 = 136.

Since 136 is divisible by 17, 986 is also divisible by 17.

For large numbers, you should apply the above-explained Rule number 2. If you are unsure about the result being a multiple of 17, repeat the process with the resultant number and keep doing this until the resultant is 0 or a multiple of 17 or the number 17 itself.

Let’s understand this theory with an example. 

We want to know if 15317 is divisible by 17 or not.

Applying rule 2– 

Subtract the last two digits (17) from two times the rest of the number (153). If that result is either a zero or a number divisible by 17, then the given number is divisible by 17.

We get, 153 × 2 – 17 = 306 – 17 = 289

We are not sure if 289 is divisible by 17. So we apply the rule again for 289. 

Now we get 89 – 2 × 2 = 89 – 4 = 85, which is a multiple of 17. 

Since 85 is divisible by 17. Therefore, 15317 is also divisible by 17.

Example

1. Check if 3978 is divisible by 17?

Applying rule 2 of the divisibility test of 17- 

Subtract the last two digits (78) from two times the rest of the number (39). If that result is either a zero or a number divisible by 17, then the given number is divisible by 17.

We get, 39 × 2 – 78 = 78 – 78 = 0. 

Since the result is 0, therefore, 3978 is divisible by 17.

2. Is 876 divisible by 17?

Applying rule 1 of the divisibility test of 17- 

Multiply the last digit by 5 and subtract that from the rest of the number. If that result is either a zero or a number divisible by 17, we can confirm that the number is divisible by 17.

We get,  87 – (6 x 5) = 57. Since 57 is not divisible by 17. Therefore,  876 is also not divisible by 17.