A number is said to be divisible by another number if the remainder is zero. Divisibility rules or divisibility tests are a set of general rules that are often used to determine whether or not a number is completely divisible by another number. In this article, some divisibility tests are given with many examples .

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**Divisibility by powers of 2 and 5**

Let any integer , then is divisible by if the integer made up of the last digits of are divisible by .

Suppose we have to determine whether an integer is divisible by 2, we only need to examine its last digit for divisibility by 2. To determine whether is divisible by 4 which is equal to , we only need to check the integer made up of the last two digits of for divisibility by 4. Similarly to determine whether is divisible by 8 which is equal to , we only need to check the integer made up of the last three digits of for divisibility by 8 and so on.

Example. Let. we see that

(i) is divisible by 2 since its last digit 8 is divisible by 2.

(ii) is divisible by since its last 2 digit 48 is divisible by 4.

(iii) is divisible by since its last 3 digit 048 is divisible by 8.

(iv) is divisible by since its last 4 digit 8048 is divisible by 16.

(v) is not divisible by since its last 5 digit 88048 is not divisible by 32.

Divisibility tests for powers of 5 are same to those for powers of 2. We only need to check the integer made up of the last digits of to determine whether is divisible by .

Example. Let n= 15535375. we see that

(i) is divisible by 5 since its last digit 5 is divisible by 5

(ii) is divisible by since its last 2 digit 75 is divisible by 25.

(iii) is divisible by since its last 3 digit 375 is divisible by 125.

(iv) is not divisible by since its last 4 digits 5375 is not divisible by 625.

** Divisibility test of 3 and 9**

To see whether is divisible by 3, or by 9, we only need to check whether the sum of the digits of is divisible by 3, or by 9.

Example : Let = 4127835. Then, the sum of the digits of is 4+ 1 +2+ 7 + 8 + 3 + 5=30. Since 30 is divisible by 3 then the given number is also divisible by 3. But 30 is not divisible by 9 , so the given number is not divisible by 9.

**Divisibility tests of 7 **

**(i) **Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7.

Example : Check whether 784 is divisible by 7.

**Solution :**In the given number 784, twice the digit in one’s place is = 2 ⋅ 4= 8

The number formed by the digits except the digit in one’s place is= 78

The difference between twice the digit in one’s place and the number formed by the other digits is = 78 – 8= 70. Since 70 is divisible by 7. So, the given number 784 is divisible by 7.

(ii) A 12-year old Nigerian boy, Chika Ofili, made history this year after his new discovery in the field of mathematics. He was awarded at the **TruLittle Hero Awards** for discovering the new divisibility test of 7, popularly called as Chika’s Test.

**Chika’s Test**

It is similar to the previously foregoing rule, but it is simpler and faster than it.

**Step 1:** Separate the last digit of the number.

**Step 2:** Multiply the last digit by 5 and add it to the remaining number.

**Step 3:** Repeat the steps unless you get a number within 0-70.

**Step 4:** If the result is divisible by 7, the number you started with is also divisible by 7.

Example: Let’s check divisibility of 23576 by 7

Sol. 2357+6×5= 2387

238+7×5=273

27+3×5=42.

Since 42 is divisible by 7, hence the given number is divisible by 7.

**Divisibility tests of 11**

**(i)** **Using alternating sums: **The given integer is divisible by 11 , if and only if the integer formed by alternately adding and subtracting the digits, is divisible by 11.

Example : . We see that 723160823 is divisible by 11, since alternately adding and subtracting its digits yields 3-2+8-0+6-1+3-2+7=22 which is divisible by 11. On the other hand, 33678924 is not divisible by 11.

since 4-2+9-8+7-6+3 – 3 =4 is not divisible by 11.

**(ii) Using pairs of digits **

step1. Write down the number.

step 2. Divide the digits in pairs from right to left. (The last digit on the left might be alone)

step 3. Add the numbers together .

step 4. Check whether the answer is divisible by 11. If it is, then the original number is divisible by 11 as well

Example. Let’s check = 17957 is divisible by 11 or not.

Divide the digits in pairs from right to left

1 | 79 | 57 |

on adding the numbers together we get 1+79+57= 137. Now check whether the answer is divisible by 11.

On repeating the same steps to check whether 137 is divisible by 11 or not , we get 1+37=38 . Since 38 is not divisible by 11, then 137 is as well. Since 137 is not divisible by 11, our original number 17957 is also not divisible by 11.

**Divisibility by 7,11 and 13 **

Step 1. Write the given number.

Step 2 . Make blocks of three digits from right to left.

Step 3. Now find alternating sum and difference of blocks of three digits.

step 4 : Check whether the answer is divisible by 7,11 and 13. If it is, then the original number is divisible by 7, 11 and 13 as well

For example : (i) Let n = 59358208. Since the alternating sum and difference of the integers formed from blocks of three digits, 208 – 358 + 59 = -91, is divisible by 7 and 13, but not by 11. We see that n is divisible by 7 and 13 but n is not divisible by 11.

(ii)Let n= 525120596. Since the alternating sum and difference of the integers formed from blocks of three digits is 596 – 120 + 525 = 1001. Since 1001 is divisible by 7, 11 and 13. We see that n is also divisible by 7 , 11 and 13.

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