## Divisibility Rules

Divisibility rules are one of the significant topics in math. We use divisibility rules almost every day and students must get to practice them effectively. Instead of teaching divisibility rules as a separate unit, they should be introduced and then referred to again and again in any applicable situation throughout the year.

Why learn divisibility rules?

• Helps in solving division problems.
• Helps in the prime factorization
• Helps in finding HCF

What are the divisibility rules?

In math, a number is said to be divisible by another number if the remainder is 0. Divisibility rules are a set of general rules that are often used to determine whether or not a number is evenly divisible by another number.

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

2: The last digit is even (0,2,4,6,8) Eg: 216 – Yes it is divisible, 127- No it is divisible

3:The sum of the digits is divisible by 3 Eg: 351 (3+5+1=9, and 9÷3 = 3) Yes
218 (2+1+8=11, and 11÷3 = 3 2⁄3) No 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

Eg: 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 a result is still a whole number. 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

5:The last digit is 0 or 5 Eg: 175 Yes, 809 No

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

8: The last three digits are divisible by 8 Eg: 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

9: The sum of the digits is divisible by 9
Eg: 1629 (1+6+2+9=18, and again, 1+8=9) Yes 2013 (2+0+1+3=6) No

10: The number ends in 0 Eg: 220 Yes, 221 No

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.
Eg: 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

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

Teaching divisibility rules through stories: