# 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

- Double the last digit (7*2=14)
- Subtract the product from step 1 and subtract from the original number without the last digit. (175-14=161)
- 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.

### Dan

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