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Nitya Nigam You are probably aware of some quick tricks to tell whether a large number is divisible by a smaller one - numbers ending in 0 or 5 are divisible by 5, numbers ending in 0, 2, 4, 6 or 8 are divisible by 2, and so on. But you may not be aware of quick tricks to tell whether a large number is divisible by 7 or 11. In this article, we'll outline and give examples for the divisibility rules from 2 to 12.
2: must end in 0, 2, 4, 6 or 8 198754 --> ends in 4, so is divisible by 2 (= 2 x 99377) 178725 --> ends in 5, so is not divisible by 2 3: sum of digits must be divisible by 3 48726 --> 4 + 8 + 7 + 2 + 6 = 27 which is divisible by 3, so it is divisible by 3 (= 3 x 16242) 92863 --> 9 + 2 + 8 + 6 + 3 = 28 which is not divisible by 3, so it is not divisible by 3 4: last 2 digits must be divisible by 4 987064 --> 64 is divisible by 4, so it is divisible by 4 (= 4 x 246766) 469254 --> 54 is not divisible by 4, so it is not divisible by 4 5: must end in 0 or 5 24975 --> ends in 5, so is divisible by 5 (= 5 x 4995) 82751 --> ends in 1, so is not divisible by 5 6: must end in 0, 2, 4, 6 or 8 and the sum of digits must be divisible by 3 98274 --> ends in 4 and 9 + 8 + 2 + 7 + 4 = 30 which is divisible by 3, so it is divisible by 6 (= 6 x 16379) 72956 --> ends in 6 but 7 + 2 + 9 + 5 + 6 = 29 which is not divisible by 3, so it is not divisible by 6 28683 --> 2 + 8 + 6 + 8 + 3 = 27 which is divisible by 3 but ends in 3, so it is not divisible by 6 92875 --> ends in 5 and 9 + 2 + 8 + 7 + 5 = 31 which is not divisible by 3, so it is not divisible by 6 7: double the last digit and subtract it from the number made by the remaining digits; result must be divisible by 7 (repeatable) 29862 --> 2x2 = 4; 2986 - 4 = 2982; 2x2 = 4; 298 - 4 = 294; 4x2 = 8; 29 - 8 = 21 which is divisible by 7, so it is divisible by 7 (= 7 x 4266) 55497 --> 7x2 = 14; 5549 - 14 = 5535; 5x2 = 10; 553 - 10 = 543; 3x2 = 6; 54 - 6 = 48 which is not divisible by 7, so it is not divisible by 7 8: last 3 digits must be divisible by 8 98272 --> 272 is divisible by 8 (= 8 x 34), so it is divisible by 8 (= 8 x 12284) 27594 --> 594 is not divisible by 8, so it is not divisible by 8 9: sum of digits must be divisible by 9 28746 --> 2 + 8 + 7 + 4 + 6 = 27 which is divisible by 9, so it is divisible by 9 (= 9 x 3194) 92875 --> 9 + 2 + 8 + 7 + 5 = 31 which is not divisible by 9, so it is not divisible by 9 10: must end in 0 29850 --> ends in 0, so is divisible by 10 (= 10 x 2985) 92875 --> ends in 5, so is not divisible by 10 11: add and subtract alternating digits; result must be divisible by 11 80256 --> 8 - 0 + 2 - 5 + 6 = 11 which is divisible by 11, so it is divisible by 11 (= 11 x 7296) 92875 --> 9 - 2 + 8 - 7 + 5 = 13 which is not divisible by 11, so it is not divisible by 11 12: last 2 digits must be divisible by 4 and the sum of digits must be divisible by 3 35820 --> 20 is divisible by 4 and 3 + 5 + 8 + 2 + 0 = 18 which is divisible by 3, so it is divisible by 12 92876 --> 76 is divisible by 4 but 9 + 2 + 8 + 7 + 6 = 32 which is not divisible by 3, so it is not divisible by 12 28475 --> 2 + 8 + 4 + 8 + 5 = 27 which is divisible by 3 but 75 is not divisible by 4, so it is not divisible by 12 97654 --> 54 is not divisible by 4 and 9 + 7 + 6 + 5 + 4 = 31 which is not divisible by 3, so it is not divisible by 12 Let us know in the comments if you found this article useful, and what mental maths tricks you'd like to see next!
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