Rules of Divisibility
A number is divisible by 3 if the sum of its digits is divisible by 3.
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 = 9 and we all know that 9 ÷ 3 = 1 so this 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.
Rule: A number is divisible by 4 if the number's last two digits are divisible by 4.
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 divisible 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 . Any multiple of 100 is divisible by four! Whether you're talking about 300, 700, 1000, 1100, 123,00-- All of these multiples of 100 are divisible by 4, which means that all that we ever have to worry about is the last two digits!
Since 6 is a multiple of 2 and 3, the rules for divisibility by 6 are a combination of the rule for 2 and the rule for 3. Rule A number is divisible by 6 if it is even and if the sum of its digits is divisible by 3.
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 (divisible 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.
Rule Divisibility by 11: A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.(This abstract and confusing sounding rule is much clearer with a few examples)
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
Rule A number passes the test for 8 if the last three digits form a number is divisible 8.
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!
Rule A number is divisible by 9 if the sum of the digits are evenly divisible 9.
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!
