Divisibility Rules and Tests

Keep this in mind before you buy graphing calculator: Even many of the best graphing calculators like the TI-89 lose their accuracy when they start dividing large numbers such as 12,347,496,132. Try to divide that large number by 11. If your calculator outputs that 12,347,496,132 is divisible by 11, it IS WRONG and should read the rule at the bottom of this page! (Look at the last example at the bottom for details) When you're dealing with exceedingly large numbers, you should rely, whenever possible, on the rules on this page rather than a scientific or a graphing calculator.

So, you may be asking yourself why is this page's calculator more accurate than that graphing calculator you bought?
• The answer is : the rules explained on this page are actually used in computing the results. .

1 + 6 + 4 + 0 = 11

11 is not divisible by 9.

Therefore 1,640 is not divisible by 9

1,640 is not evenly divisible by 9 because there is a remainder of 2.
9 • 182 is 1,638 which leaves a remainder of 2

i.e. 1,640 = 9 • 182 + 2 = 1,638 + 2

Examples of numbers that are even and therefore pass this divisibility test
• 2
• 0
• 4
• -2
• -312
• 31,102

Examples of numbers that are do not pass this divisibility test because they are not even.
• 3
• -103
• 1.50
• 221

 Divisibility by 3 Rule  

Examples of numbers that are divisible by 3
• 12 1 + 2 = 3  And 3 is divisible by 3 so the number 12 is also divisible by 3.
• 36 3 + 6 = 9 And 9 is divisible by 3 which means that 36 is also.
• 102 1 + 0 + 2 = 3
• 100,002,000 = 1 +0 +0 +0 +0 +2 +0 + 0 +0  = 3 so this very large number passes this divisibility test.
• -36 = -3 + 6 = 3 and we all know that 3 ÷3 = 1 so this negative number satisfies the rule and is evenly divided by 3!

Examples of numbers that do not pass this test  14
• 1+4 = 5 and since 5 is not divisible by 3, so 14 is also not.
• 124 1 + 2+ 4 = 7 which is no good, does not work.
• 100,002,001 = 1 +0 +0 +0 +0 +2 +0 + 0 + 1= 4 so this very large also does not pass this divisibility test.

 Divisibility by 4 Rule 

Examples of numbers that are divisible by 4
• 112 since the last two digits, 12, are divisible by 4, the number 112 is satisfies this rule and is also divisible by 4.
• 10,948 the last two digits, 48, are divisible by 4. Therefore, the whole number is also.
• 100,002,088 = 88. Yep, this satisfies rule because 88 is divisible by 4!
• -12,036 = 36 and 36 is evenly divided by 4, so -12,036 passes the test!

Examples of numbers that are do not pass this divisibility test
• 113 since the last two digits, 13, are not divisible by 4, the whole number does not pass this divisibility test.
• 10,941 the last two digits, 41, are not de visible by 4. Therefore, the whole number does not satisfy the rule for 4.
• 100,002,014 =  14 and 14 is no good, does not work.
• -1,011 = 11 so 1,011 fails this test .

Examples of numbers that are divisible by 5 and satisfy this rule
• 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
• 12,340

Examples of numbers that fail this divisibility test.
• 17 since the last digit is 7, 17 does not satisfy this rule and is not divisible by 5
• 118 since the last digit is 8, 118 does not satisfy this rule and is not divisible by 5
• -311 Since the last digit is 1, 311 does not satisfy the rule for 5
• -101
• 12,103

Examples of numbers that are divisible by 6
• 12   satisfies both conditions:
  • 1) 12 is even
  • 2) the sum of its digits (1+2 =3)  is divisible by 3. Therefore, 12 passes this test.
• 114 satisfies both conditions
  • 1) 1+1+4 = 6 which is divisible by 3
  • 2) 114 is even
• 241,122 This passes the test because it's even and the sum of its digits can be evenly divided by 3.

Examples of numbers that are do not pass this divisibility test
• 207 Fails the test since it's not even. We don't even have to see whether the second condition is satisfied since both conditions must be satisfied to pass this test. If only one of the two conditions (divisble by 2 and by 3) are not met, then the number does not satisfy the rule for 6.
• 241,124 Although this number is even, the sum of its digits are not evenly divided by 6 so this fails the test.

Examples of numbers that satisfy this rule and are divisible by 8
• 9,640 640 ÷ 8 = 80 so the whole number, 9,640, is divisible by 8
• 77, 184 184 ÷ 8 = 23 so  77,184 passes this divisibility test.
• 67, 536 536 is divisible by 8 ( 536 ÷ 8 = 67) so 67,536, is also.
• -30 640 ÷ 8 = 80 so the whole number, 9640, passes this test.
• 20,233,322,496 Well, maybe you were wondering if this divisibility rule was really helpful or not. Once you get a giant number like 20,233,322,496, you start to realize what a nice trick this is to have up your sleeve! All you have to do is divide 496 by 8 to learn that the entire number is divisible by 8.
• - 316,145,664 664 passes this divisibility test.

Examples of numbers that are do not pass this divisibility test
• 9,801 since 801 is not divisible by 8, 9,801 is not.
• 234,516 Nope, no good. 516 is not evenly divided by 8 so the whole number fails the test!
• -32,344,588 588 does not work, so -32,344,588 does not satisfy the rule for 8!

Examples of numbers that satisfy this rule and are divisible by 9
• 4,518 4+5+1+8=18 and since 18 ÷9 = 2 , the whole number, 4,518, is divisible by 9
• 7,209 7+2+0+9 = 18, and by the same logic of the prior example, 7,209 passes this divisibility test.
• 6,993 ,6993 is divisible by 9(6+9+9+3 = 27 & 27 ÷ 9 = 3) so 6,993 satisfies the rule for 9.
• 10,006,470 Well, maybe you were wondering if this divisibility trick was really helpful or not. Once you get a giant number like 10,006,470, you start to realize what a nice trick this is to have up your sleeve! All you have to do is add the digits (1+ 6+4+7 = 18) to quickly see that the entire number is divisible by 9 (18÷9 = 2).

Examples of numbers that are do not pass this divisibility test
• 29 2+9 =11. Since 11 is not divisible by 9, 29 is not either.
• 6,992 Nope, no good. 6+9+9+2 =26 which is not evenly divided by 9 so the whole number fails the test!

Examples that pass this test
• 100
• 110
• -110
• 1,320,320.

Examples of numbers that are do not pass this divisibility test
• 91,801 last digit is not zero, so this does not work.
• 234,516 Nope, this number does not satisfy the rule for 10.
• -32,344,508 Again, it all comes down to that last digit which just has to be zero!

Examples of numbers that satisfy this rule
• 946 (9+6) - 4 = 11 which is, of course, evenly divided by 11 so 946 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 does indeed work. In case you found this last bit confusing, remember that any number evenly divides 0. Think about it, how many 11's are there in 0? None, right. Well that means that 11 divides zero, zero times!
• 119,777,658 (1+ 9 + 7 + 6 + 8) - (1+ 7 + 7 +5) = 31 - 20 = 11

Examples of numbers that are do not pass this divisibility test
• 947 (9+7) - 4 = 12 which is not divisible by 11
• 10,823 (1+8+3) - (0+2) = 12- 2 =10. No, no good. This one fails!
• 35, 784 = (3 + 7 + 4) - (5+8) = 14 - 13 = 1.  No, does not satisfy the rule for 11!
• 12,347, 496, 132 = (1+3+7+9+3) - (2 + 4 +4 + 6 + 3)= 23- 19 = 4