Rules of Divisibility

Too many students arrive at high school without the basic math fundamental skills.  The rules of divisibility is one of the critical skills that is crucial to being able to understand conceptually division and fractions.  Below is listed the rules of divisibility for the number 1-12.

1 

​Any integer (not a fraction) is divisible by 1

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2  

​The last digit is even (0,2,4,6,8)
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128  Yes
129  No

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​3  

​The sum of the digits is divisible by 3

381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, and 10÷3 = 3 1/3) No
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This rule can be repeated when needed:
99996 (9+9+9+9+6 = 42, then 4+2=6) Yes

​4  

​The last 2 digits are divisible by 4

1312 is (12÷4=3) Yes
7019 is not (19÷4=4 3/4) No

A quick check (useful for small numbers) is to halve the number twice and the result is still a whole number.
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12/2 = 6, 6/2 = 3, 3 is a whole number. Yes
30/2 = 15, 15/2 = 7.5 which is not a whole number. No

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​5 

The last digit is 0 or 5
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175  Yes
809  No

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​6  

​Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)

114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes
308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No

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​7  

​Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7. (We can apply this rule to that answer again)

672 (Double 2 is 4, 67-4=63, and 63÷7=9) Yes
105 (Double 5 is 10, 10-10=0, and 0 is divisible by 7) Yes
905 (Double 5 is 10, 90-10=80, and 80÷7=11 3/7) No

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​8  

​The last three digits are divisible by 8

109816 (816÷8=102) Yes
216302 (302÷8=37 3/4) No

A quick check is to halve three times and the result is still a whole number:

816/2 = 408, 408/2 = 204, 204/2 = 102 Yes
302/2 = 151, 151/2 = 75.5 No

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​9  

​The sum of the digits is divisible by 9

(Note: This rule can be repeated when needed)
1629 (1+6+2+9=18, and again, 1+8=9) Yes
2013 (2+0+1+3=6) No

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​10  

​The number ends in 0

220  Yes
221  No

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​11  

​Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.
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1364 (+1−3+6−4 = 0) Yes
913 (+9−1+3 = 11) Yes
3729 (+3−7+2−9 = −11) Yes
987 (+9−8+7 = 8) No

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12 

​​The number is divisible by both 3 and 4 (it passes both the 3 rule and 4 rule above)

648 
(By 3? 6+4+8=18 and 18÷3=6 Yes) 
(By 4? 48÷4=12 Yes) 
Both pass, so Yes

524 

(By 3? 5+2+4=11, 11÷3= 3 2/3 No) 
(Don't need to check by 4) No