Test For Divisibilty

Test for divisibility are short ways to determine the factors of a natural number without doing the actual calculation.

Divisibility Test for 2:

All even numbers are divisible by 2.

Example:  4, 24, 98, 28890

Divisibility Test for 3:

A number is divisible by 3 if the sum of the digits of the number is divisible by 3. Like 190191 is divisible by 3 because the sum of the digits is 1+9+0+1+9+1=21 and 21 is divisible by 3

Sample problem:

Find the smallest and largest value of a in 781a01 so that the number is divisible by 3.

Solution:

Sum of digits is 7+9+1+a+0+1=a+17. To get the smallest value start testing from 0 but if a=0. 17 is not divisible by 3.Test 1, 18 is divisible by 3. So smallest vale of a=1. To get the largest possible value of a, start counting from 9. After testing the largest value of a=7.

Divisibility Test for 4

A number is divisible by  4 if the the last two digit of the number is divisible by 4. 8934239012344 is divisible by because 44 is divisible by 4.

Sample problem:

For what value/values of a so that the number 89769793048475873a2 is divisible by 4?

Solution:

Just focus your attention to last two digit which is a2. test each number from 0-9.

Possible value of 2 are: {1,3,5,7,9}

Divisibility Test for 5:

A number is divisible by 5 if the last digit of the number is either 0 or 5. 3645348586450 and 883943245 are both divisible by 5.

Sample problem:

For what value(s) of a in 998435a0030 such that the number is divisible by and 5?

Solution:

the last digit of the number is 0. And as mention above any value of a will not affect the property of the number. So possible value of a={1,2,3,4,5,6,7,8,9,0}

Divisibility Test for 6:

the number is divisible by 6 if the number if the number is even and divisible by 3. The number 23142 is divisible by 6 since it is even and the sum of the digits is divisible by 3.

Sample problem:

A number 39499a10 is divisible by 6. What is the smallest value of a?

Solution:

We need to achieve 2 conditions. Even number and divisible by 3. The number is already even so the number must be divisible by 3.

sum of digits= 3+9+4+9+9+a+1+0=35+a. Smallest value of a=1.

Divisibility Test for 7:

To test if the number is divisible by 7. We need to follow some steps. Is 1757 divisible by 7? To answer this question we will do the following steps.

STEPS

1. Double the last digit (7*2=14)
2. Subtract the product from step 1 and subtract from the original number without the last digit. (175-14=161)
3. If the difference is divisible by 7 then the number is divisible by 7. If the number is big repeat step 1 and 2.

Repeating the process it will end up to 16-2=14 which is divisible by 7. Therefore 1757  is divisible by 7.

Divisibility Test for 8:

If the last three digit of the number is divisible by 8 the number is divisible by 8. So we need to perform long division but will only take the last three digit of the dividend. 97974578645968 is divisible by 8 because 968 is a multiple of 8.

Sample problem:

What is the value of a in 8349487512a so that the number is divisible by 8?

Solution:

12a must be divisible by 8. 12a is also 120+a. 120 is multiple of 8. So a=8 is the only solution.

Divisibility Test for 9:

A number is divisible by 9 if the sum of the digits of a number is also divisible by 9918 is divisible by 9 because the sum of digits is (9+9+1+8=27) is divisble by 9.

Sample problem:

For what value of a in 84a09 is the number divisible by 9?

 Solution:

The sum of the digits is 8+4+a+0+9=a+21. So a=6 for the digit to be divisible by 9.