## Sunday, June 27, 2010

### Simplified DIVISIBILITY TESTS for 9, 3, 4 and 8

Simplified DIVISIBILITY TESTS for 9, 3, 4 and 8

http://gadremathematics.blogspot.com/

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.

21656 56 and hence the number is divisible by 8.

82686 86 and hence the number is NOT divsible by 8.

51744 44 + 4 = 48 and hence the number is divisible by 8.

89324 24 + 4 = 28 and hence the number is NOT divsible by 8.