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Divisibility Rules

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Divisibility by 2 Rule


Almost everyone is familiar with this rule which states that any even number can be divided by 2. Even numbers are multiples of 2. A number is even if ends in 0,2,4,6, or 8.
Ex.
  • 2
  • 0
  • 4
  • -2
  • -312
  • 31,102

Divisibility by 3 Rule


Add up all the digits in the number. Find out what the sum is. If the sum is divisible by 3, so is the number
Ex.
12→1 + 2 = 3 And 3 is divisible by 3 so the number 12 is also divisible by 3.

Divisibility by 4 Rule


A number is divisible by 4 if the number's last two digits are divisible by 4.
Ex.
358912 ends in 12 which is divisible by 4, thus so is 358912

Divisibility by 5 Rule


A number is divisible by 5 if the its last digit is a 0 or 5.
Ex.
  • 10 → since the last digit is 0, 10 satisfies this rule and is divisible by 5
  • 15 → since the last digit is 5, 15 satisfies this rule and is divisible by 5
  • 45
  • -30
  • 55
  • -105

Divisibility by 6 Rule


If the Number is divisible by 2 and 3 it is divisible by 6 also.
Ex. 
 114 → satisfies both conditions
  • 1) 1+1+4 = 6 which is divisible by 3
  • 2) 114 is even so divisible by 2
  • Hence 114 is divisible by 6


Divisibility by 7 Rule


Test - 1
Take the last digit in a number. Double and subtract the last digit in your number from the rest of the digits. Repeat the process for larger numbers.
Ex
357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

Test - 2
Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary Add the products. If the sum is divisible by 7 - so is your number.
Ex
Example: Is 2016 divisible by 7? 6(1) + 1(3) + 0(2) + 2(6) = 21 21 is divisible by 7 and we can now say that 2016 is also divisible by 7

Divisibility by 8 Rule


A number passes the test for 8 if the last three digits form a number is divisible 8.
Ex.
6008 - The last 3 digits are divisible by 8, therefore, so is 6008.

Divisibility by 9 Rule


A number is divisible by 9 if the sum of the digits are evenly divisible 9.
Ex.
43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!

Divisibility by 10 Rule


A number passes the test for 10 if its final digit is 0.
Ex.
  • 100
  • 110
  • -110
  • 1,320,320.

Divisibility by 11 Rule


A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.
Ex.
  • 682 → (6+2) - 8 = 0 which is, of course, evenly divided by 11 so 682 passes this divisibility test
  • 10,813 → (1+8+3) - (0+1) = 12-1 =11. Yes, this satisfies the rule for 11!
  • 25, 784 = → (2+ 7 + 4) - (5+8) = 13 - 13 =0 . Yes this also satisfies the rule for 11!



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