Arithmetic - Basics 3

Friday, April 28, 2006

Suppose you want to find if the number 9475724 is divisible by 4, what will you do? Will you divide the whole number and check if it is divisible or not? If yes, then wait, there are aesthetic methods to do so.

Divisibility tests:

2: If the units place of the number is even, the number is divisible by 2, e.g. 2, 4, 6, 8 and 0.

3: If the sum total of digits of the number is divisible by 3 then the number is divisible by three, e.g. 633 = 6 + 3 + 3 = 12, which is divisible by 3, hence the number 633 is also divisible by 3.

4: If the last 2 digits of the number is divisible by 4, the number will be divisible by 4.

5: If the units place of the number is divisible by 5, means should be 0 or 5, the number will be divisible by 5.

6: If the units place of the number is divisible by 2 and 3 both then the number is divisible by 6.

7: For 7 the method is little different, let us look at an example, suppose you want to find if 6764 is divisible by 7 or not. We have to start from left to right, i.e from units place, take 4 multiply it with 2 and subtract it with remaining i.e 676, then whatever the answer take its unit place multiply with 2 and subtract with remaining, go on doing until you are left with a number that you can easily identify as a multiple of 7 or not.
Step 1) 676 - 4 x 2 = 676 - 8 = 668
Step 2) 66 - 8 x 2 = 66 - 16 = 50, stop here, you know 50 is not a multiple of 7, so we can peacefully say that the number 6764 is not divisible by 7. Later we will see why we took 2 as a multiple.

8: If the last 3 digits of the number is divisible by 8, the number is also divisible by 8, also if the last 3 digits are 0's. Actually there is a easier method, see if the last 2 digits are divisible by 4 and the hundred's place digit is odd, then the number is divisible by 8 or if the last 2 digits are divisible by 8 and the hundred's place digit is even, then the number will be divisible by 8.

9: If the sum total of digits of the number is divisible by 9 then the number is divisible by nine, similar to divisibility rule for 3.

11: If the sum difference of even numbers and odd numbers is divisible by 11 or is 0, then the number is divisible by 11, example take 13574.
Step 1) 1 + 5 + 4 = 10, All the digits in the odd places.
Step 2) 3 + 7 = 10, all the digits in the even places
Step 3) 10 - 10 = 0, their difference, hence the number is divisible by 11.

12: If the number is divisible by 4 and 3 then the number is divisible by 12. Wonder why we check for 4 and 3? Because the factors of 12 are 4 and 3 and they are co-primes.

Some Observations:
1) If z divides both x and y, then (x + y) and (x - y) are divisible by z.
Example: 2 divides both 4 and 12, so (4 + 12) and (4 - 12) will both be divisible by 2.

2) Sum of 5 consecutive whole numbers is always divisible by 5.
Example: 1 + 2 + 3 + 4 + 5 = 15, hence divisible by 5.

3) The product of three consecutive numbers, if the first number is even, the result will always be divisible by 24. Why? Because the above numbers will always have factors 8 and 3.
Example: 2 x 3 x 4 = 24 or 4 x 5 x 6 = 120, both the numbers, 24 and 120 are divisible by 24.

4) The product of three consecutive numbers, if the first number is odd, then the result will always be divisible by 6. Why? Because the above numbers will always have factors 2 and 3.
Example: 3 x 4 x 5 = 60 or 5 x 6 x 7 = 210, both the numbers, 60 and 210 is divisible by 6.

5) Difference between a number and the number formed by writing its digits in reverse order is divisible by 9.
Example: 4321 - 1234 = 3087, which is divisible by 9. (Remember, if the sum of the digits is divisible by 9, then the number is divisible by 9.)

6) Any number (10^n) - 1 is divisible by 9.
Example: 10^3 - 1 = 1000 - 1 = 999