Dividing Evenly

Background

Are there ways to tell if a number is evenly divisible by smaller numbers?

Question

What is the way to know if a number is divisible by the numbers 2 - 9?

Answer

There are methods - see below:

Analysis

When we're looking at a large number and we're looking at trying to find factors, it's helpful to know if that large number is divisible evenly by smaller numbers. We can run through some rules on some of the more common factors we'll run into:

Divisible by:

• 2 - If the number is even

• 222 is even and divisible by 2
• 223 is odd and not divisible by 2 

• 3 - If the sum of the digits of the number are divisible by 3

• 222 - we sum the digits 2 + 2 + 2 = 6. 6 is divisible by 3, so 222 is divisible by 3
• 223 - we sum the digits 2 + 2 + 3 = 7. 7 is not divisible by 3, so neither is 223 

• 4 - If the last two digits of the number are divisible by 4

• 224 - the last two digits are 24, which is divisible by 4, so 224 is divisible by 4
• 225 - the last two digits are 25, which is not divisible by 4, so 225 is not divisible by 4 

• 5 - If the last digit of the number is either a 5 or a 0

• 225 ends in 5, so 225 is divisible by 5
• 226 does not end in 0 or 5, so 226 is not divisible by 5 

• 6 - A combo of divisible by 2 and 3 - even and the digits sum to 3

• 222 is even and the digits sum to 6, so 222 is divisible by 6
• 223 is not even, so 223 is not divisible by 6
• 224 is even but the sum of the digits is 2 + 2 + 4 = 8 which is not divisible by 6, so 224 is not divisible by 6 

• 7 - Double the last digit and subtract from the remaining digits. If that difference is divisible by 7

• 224 - double the 4, which is 8. 22 - 8 = 14. 14 is divisible by 7, so 224 is divisible by 7
• 225 - double the 5, which is 10. 22 - 10 = 12. 12 is not divisible by 7, so 225 is not divisible by 7 

• 8 - The last 3 digits are divisible by 8

• 777,888 - the last 3 digits are 888 and are divisible by 8. Therefore 777,888 is divisible by 8
• 888,777 - the last 3 digits are 777 but are not divisible by 8. Therefore 888,777 is not divisible by 8 

• 9 - The sum of the digits is divisible by 9

• 8190 - The sum of the digits is 8 + 1 + 9 + 0 = 18. 18 is divisible by 9, so 8190 is also
• 9991 - The sum of the digits is 9 + 9 + 9 + 1 = 28. 28 is not divisible by 9, so neither is 9991 

Vocabulary used:

For more information check out these links (comment to add your favourite link):

https://www.quickanddirtytips.com/education/math/how-to-tell-if-a-number-is-divisible-by-2-or-3
https://www.quickanddirtytips.com/education/math/how-to-tell-if-a-number-is-divisible-by-4-5-or-6
https://www.quickanddirtytips.com/education/math/how-to-tell-if-a-number-is-divisible-by-7-8-or-9

Where might you have come from?

Fact-orials Index

Operations:

• Division

Associated Operations:

• Factors and Factoring

Where might we go?

Associated Operations:

• Remainders and Modulo
• Long Division