Divisibility Test - Mathematics Form 1 Notes

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Introduction

  • divisibility Test is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.


Divisibility Test for 2

  • A number is divisible by 2 If and only if its ones digit is divisible by 2 (i.e. the ones digit is either 0, 2, 4, 6, or 8, or if the last digit is an even number)

Ex: 92,659,354,236 is divisible by 2 since the ones digit is a 6

642 is divisible by two because it ends in a two, which makes it an even number



Divisibility Test for 3

  • A number is divisible by 3 if and only if the sum of its digits is divisible by 3

Ex: 954 is divisible by 3 since 9 + 5 + 4 = 18 is divisible by 3

423 is divisible by three because 4 + 2 + 3 = 9. Since nine is a multiple of three (or is divisible by three), then 423 is divisible by three



Divisibility Test for 4

  • A number is divisible by 4 if and only if the last 2 digits taken as a number itself is divisible by 4

Ex: 5632 is divisible by 4 since 32 is divisible by 4

338 and 2738 38 is not a multiple of 4. 338 and 2738 are not divisible by 4.

272 and 3172 72 is a multiple of 4. 272 and 3172 are divisible by 4.



Divisibility Test for 5

  • A number is divisible by 5 if and only if the ones digit is either 0 or 5.

435 is divisible by five because it ends in a five

 



Divisibility Test for 6

  • A number is divisible by 6 if and only if it is divisible by both 2 and 3

Ex: 954 is divisible by 6 since it is divisible by both 2 (the ones digit is a 4) and 3

222 is divisible by six because it is even, so it is divisible by two and its digits add up to six, which makes it divisible by three



Divisibility Test for 7

  • A number is divisible by 7 if and only if the difference between 2 times the ones digit and the number formed by its remaining digits is divisible by 7

Ex: 133 is divisible by 7 since 2 x 3 = 6 and 13 – 6 = 7 is divisible by 7

441 is divisible by 7 since 2 x 1 = 2 and 44 – 2 = 42 is divisible by 7



Divisibility Test for 8

  • A number is divisible by 8 if and only if the last 3 digits taken as a number itself is divisible by 8

Ex: 34,848 is divisible by 8 since 848 is divisible by 8



Divisibility Test for 9

  • A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Ex: 954 is divisible by 9 since 9 + 5 + 4 = 18 is divisible by 9

9243 is divisible by nine because the sum of the digits adds up to eighteen, which is a multiple of nine



Divisibility Test for 10

  • A number is divisible by 10 if and only if the ones/last digit is 0. 

730 is divisible by ten because it ends in zero



Divisibility Test for 11

  • A number is divisible by 11 if and only if the sum of the digits in the odd places (ones, hundreds, ten-thousands, millions, etc.) minus the sum of the digits in the even places (tens, thousands, hundredthousands, ten-millions, etc.) is 0 or divisible by 11
  • Which in other words can be written as, a number is divisible by 11 if the difference in sum of alternate digits, is 0, 11 or a multiple of 11.

363 is divisible by 11 since (3 + 3) – 6 = 0 is divisible by 11

737 is divisible by 11 since (7 + 7) – 3 = 11 is divisible by 11



Divisibility Tests Practice Questions

  1. Determine whether 644 is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11
  2. Determine whether 3120 is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11
  3. Determine whether 4725 is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11
  4. Which of the following numbers are divisible by 2, 5 and 10?
    1. 149
    2. 19400
    3. 720345
    4. 125370
    5. 3000000
  5. Check whether the numbers are divisibility by 4:
    1. 23408
    2. 100246
    3. 34972
    4. 150126
    5. 58724
    6. 19000
    7. 43938
    8. 846336
  6. In each of the following numbers without doing actual division, determine whether the first number is divisible by the second number:
    1. 3409122; 6
    2. 17218; 6
    3. 11309634; 8
    4. 515712; 8
    5. 3501804; 4
    6. 14641; 11
  7. 6 is a factor of 12066 and 49320. Is 6 a factor of 49320 + 12066 and 49320 - 12066?
  8. Is 9 a factor of the following?
    1. 394683
    2. 1872546
    3. 5172354
  9. Fill in the smallest digit to make the number divisible by:
    1. by 5 : 7164__, 32197__
    2. by 3 : 1__43, 47__05, __316
    3. by 6 : __428, 9__52, 721__
    4. by 4 : 2462__, 91__ __, 670__ 
    5. by 8 : 1232__, 59__16, 4642__
  10. Which of the two nearest numbers to 19506 are divisible by 9?
  11. Check whether the following numbers are divisible by 11
    1. 62928
    2. 1936
    3. 190817
    4. 6368
    5. 1364

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