Mathematics: Rules of Divisibility

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Rules of Divisibility: This article focuses on the Rules of Divisibility in mathematics. You can also learn the Rules of divisibility for number 2, 3, 5, 7, 9, 11, 13, 40 and 80, etc.

What is divisibility?

divisibility test is a rule for determining whether one whole number is divisible by another. It is a quick way to find factors of large numbers.

In other words, we can say that Divisibility means “when you divide one number by another the result will be a whole number

Rules of Divisibility

  1. Divisibility rules for 2: A number is divisible by 2 if it’s unit’s digit is any of 0, 2, 4, 6, 8. Ex. 84932 is divisible by 2, while 65935 is not.
  1.  Divisibility rules for 3: A number is divisible by 3, if the sum of its digits is divisible by 3. Ex.592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3. But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32, which is not divisible by 3.
  1. Divisibility rules for 4: A number is divisible by 4 if the number formed by the last two digits is divisible by 4. Ex. 892648 is divisible by 4 since the number formed by the last two digits is 48,  which is divisible by 4. But, 749282 is not divisible by 4, since the number formed by the last tv/o digits is 82, which is not divisible by 4.
  1. Divisibility rules for 5: A number is divisible by 5 if it’s unit’s digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
  1. Divisibility rules for 6: A number is divisible by 6 if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2. Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
  1. Divisibility rules for 8: A number is divisible by 8 if the number formed by the last three digits of the given number is divisible by 8. Ex. 953360 is divisible by 8 since the number formed by last three digits is 360, which is divisible by 8. But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.
  2. Divisibility rules for 9: A number is divisible by 9, if the sum of its digits is divisible by 9. Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9. But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9.
  1. Divisibility rules for 10: A number is divisible by 10 if it ends with 0. Ex. 96410, 10480 are divisible by 10, while 96375 is not.
  2. Divisibility rules for 11: A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11. Ex. The number 4832718 is divisible by 11, since : (sum of digits at odd places) – (sum of digits at even places) (8 + 7 + 3 + 4) – (1 + 2 + 8) = 11, which is divisible by 11.
  1. Divisibility rules for 12: A number is divisible by 12 if it is divisible by both 4 and 3. Ex. Consider the number 34632.
  • The number formed by last two digits is 32, which is divisible by 4,
  • (ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
  1. Divisibility rules for 14: A number is divisible by 14 if it is divisible by 2 as well as 7.
  2. Divisibility rules for 15: A number is divisible by 15 if it is divisible by both 3 and 5.
  3. Divisibility rules for 16: A number is divisible by 16 if the number formed by the last 4 digits is divisible by 16.
  • Ex.7957536 is divisible by 16 since the number formed by the last four digits is 7536, which is divisible by 16.
  1. Divisibility rules for 24: A given number is divisible by 24 if it is divisible by both 3 and 8.
  1. Divisibility rules for 40: A given number is divisible by 40 if it is divisible by both 5  and 8.
  1. Divisibility rules for 80: A given number is divisible by 80 if it is divisible by both 5 and 16.

Note :

  1. If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by PQ.
  2. If p arid q are not co-primes, then the given number need not be divisible by PQ, even when it is divisible by both p and q.
  • Ex. 36 is divisible by both 4 and 6, but it is not divisible by (4×6) = 24 since 4  and 6 are not co-primes.

Let’s see more example of Rules of Divisibility

Example 1: Determine whether 150 is divisible by 2, 3, 4, 5, 6, 9 and 10.

150 is divisible by 2 since the last digit is 0.

150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6), and 6 is divisible by 3.

150 is not divisible by 4 since 50 is not divisible by 4.

150 is divisible by 5 since the last digit is 0.

150 is divisible by 6 since it is divisible by 2 AND by 3.

150 is not divisible by 9 since the sum of the digits is 6, and 6 is not divisible by 9.

150 is divisible by 10 since the last digit is 0.

Solution: 150 is divisible by 2, 3, 5, 6, and 10.

Example 2: Determine whether 225 is divisible by 2, 3, 4, 5, 6, 9 and 10.

225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.

225 is divisible by 3 since the sum of the digits is 9, and 9 is divisible by 3.

225 is not divisible by 4 since 25 is not divisible by 4.

225 is divisible by 5 since the last digit is 5.

225 is not divisible by 6 since it is not divisible by both 2 and 3.

225 is divisible by 9 since the sum of the digits is 9, and 9 is divisible by 9.

225 is not divisible by 10 since the last digit is not 0.

Solution: 225 is divisible by 3, 5 and 9.

Example 3: Determine whether 7,168 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.

7,168 is divisible by 2 since the last digit is 8.

7,168 is not divisible by 3 since the sum of the digits is 22, and 22 is not divisible by 3.

7,168 is divisible by 4 since 168 is divisible by 4.

7,168 is not divisible by 5 since the last digit is not 0 or 5.

7,168 is not divisible by 6 since it is not divisible by both 2 and 3.

7,168 is divisible by 8 since the last 3 digits are 168, and 168 is divisible by 8.

7,168 is not divisible by 9 since the sum of the digits is 22, and 22 is not divisible by 9.

7,168 is not divisible by 10 since the last digit is not 0 or 5.

Solution: 7,168 is divisible by 2, 4 and 8.

Example 4: Determine whether 9,042 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.

9,042 is divisible by 2 since the last digit is 2.

9,042 is divisible by 3 since the sum of the digits is 15, and 15 is divisible by 3.

9,042 is not divisible by 4 since 42 is not divisible by 4.

9,042 is not divisible by 5 since the last digit is not 0 or 5.

9,042 is divisible by 6 since it is divisible by both 2 and 3.

9,042 is not divisible by 8 since the last 3 digits are 042, and 42 is not divisible by 8.

9,042 is not divisible by 9 since the sum of the digits is 15, and 15 is not divisible by 9.

9,042 is not divisible by 10 since the last digit is not 0 or 5.

Solution: 9,042 is divisible by 2, 3 and 6

Example 5: Determine whether 35,120 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.

35,120 is divisible by 2 since the last digit is 0.

35,120 is not divisible by 3 since the sum of the digits is 11, and 11 is not divisible by 3.

35,120 is divisible by 4 since 20 is divisible by 4.

35,120 is divisible by 5 since the last digit is 0.

35,120 is not divisible by 6 since it is not divisible by both 2 and 3.

35,120 is divisible by 8 since the last 3 digits are 120, and 120 is divisible by 8.

35,120 is not divisible by 9 since the sum of the digits is 11, and 11 is not divisible by 9.

35,120 is divisible by 10 since the last digit is 0.

Solution: 35,120 is divisible by 2, 4, 5, 8 and 10.

Example 6: Is the number 91 prime or composite? Use divisibility when possible to find your answer.

91 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.

91 is not divisible by 3 since the sum of the digits (9+1=10) is not divisible by 3.

91 is not evenly divisible by 4 (the remainder is 3).

91 is not divisible by 5 since the last digit is not 0 or 5.

91 is not divisible by 6 since it is not divisible by both 2 and 3.

91 divided by 7 is 13.

Solution: The number 91 is divisible by 1, 7, 13 and 91. Therefore 91 is composite since it has more than two factors.

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