__Simplified ____DIVISIBILITY TESTS for 9, 3, 4 and 8__

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**Conventional Test for 9 (or 3) –** If sum of the digits of the number is divisible by 9 (3) then number is divisible by 9 (3). The number 2754 is divisible by 9 (3) as sum of its digits = 2+7+5+4 = 18 is divisible by 9 (3).

**Here 2, 7, 5 and 4 are remainders when place value of the digits namely 2000, 700, 50 and 4 are divided by 9**. When the sum of these remainders is divisible by 9 (3), the number is divisible by 9 (3).

**In simplified test for 9, before making the sum of digits, first discard the remainders giving sum equal to 9 or multiple of 9 (“In test for 3” – sum equal to 3 or multiple of 3).**

7247583 → [~~72~~](4){7}(5){83} → Discarding sums 9,9, and 18, remainder is 0. Therefore the number is divisible by 9 (3).

72476583→[~~72~~](4){7}**6**(5){83}→ Discarding sums 9,9, and 18 Remainder is 6. The number 72476583 is not divisible by 9. It will give remainder 6 when divided by 9. But the number is divisible by 3 as the remainder 6 is divisible by 3.

**Conventional Test for 4 –**When the number formed by last two digits of a number is divisible by 4 then the number is divisible by 4.

**In simplified approach this two digit number is written as sum of numbers.**

6936 is divisible by 4 as 36 is divisible by 4.

2376 is divisible by 4 as 76 = 40 + 36 and both 40 and 36 are divisible by 4.

8992 is divisible by 4 as 92 = 40 + 40 +12 and 40, 40and 12 are all divisible by 4.

**Conventional Test for 8 –**When the number formed by last three digits of a number is divisible by 8 then the number is divisible by 8.

**In Simplified approach “last two digits” or “last two digits plus 4” are tested for 8 depending on whether digit at hundred is Even or Odd.**

(A) __Even hundreds__ are divisible by 8.Hence check only the number formed by last two digits is divisible by 8.

(B) __Odd hundreds__ give remainder 4 when divided by 8. Hence check the sum of the number formed by last two digits and remainder 4 is divisible by 8.

21**6**56 → 56 and hence the number is divisible by 8.

82**6**86 → 86 and hence the number is NOT divsible by 8.

51**7**44 → 44 + 4 = 48 and hence the number is divisible by 8.

89**3**24 → 24 + 4 = 28 and hence the number is NOT divsible by 8.