For a number to be divisible by 13, the product of its digit in the unit’s place with 4, when added to the number formed by the rest of its digits must be 0 or a multiple of 13. A divisibility rule is a useful shorthand for determining whether a given integer is divisible by a fixed divisor without performing the division, usually by inspecting and performing basic fundamental theorems on arithmetic operations on its digits. There are different divisibility tests for numbers in any radix or base. There are four different divisibility rules for 13.

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What are Divisibility Rules for 13?

Divisibility rules for 13 are a set of rules that can be used to determine whether a given number is divisible by 13 or not. The rule for divisibility of a number by 13 states that a number is divisible by 13 when its one’s place digit is multiplied by 4 and this product when added to the number formed by the rest of its digits, is either 0 or a multiple of 13.

• Divide the last digit by four.
• Add the result to the remaining truncated leading number.
• If the outcome is divisible by 13, the original number is also divisible by 13.
• Apply this rule as many times as necessary.

Example: Determine whether the number 559 is divisible by 13.

Solution: Applying the divisibility rule to the number 13,

Step 1: The number 559’s last digit is 9.

Divide ‘9’ by 4

9×4=36

Step 2: Add this number to the remaining incomplete number which is 55

36+55 = 91

Step 3: The number 91 is divisible by 13.

As a result, the number 599 is also divisible by 13

Rules of Divisibility Rules for 13 are given below.

Divisibility by 13 Rule 1

Rule 1:

Step: 1 Begin by grouping the given number into sets of three.

Step: 2 Apply the subtraction and addition operations alternately to the rightmost group of three digits and find the result.

Step: 3 If the result is 0 then, the number is divisible by 13.

Example: 1,111,500 is divisible by 13 or not

Step 1: 1,111,500 In this number, using the subtraction and addition operations alternately from the rightmost group of three digits,

we obtain, 500- 111 + 1 = 389+1 = 390

Step 2: Now, 390 divided by 13 Then we get,

390÷13=30 as the quotient and 0 as the remainder.

Step 3: As a result, 1.111,500 is divisible by 13.

Divisibility by 13 Rule 2

Rule 2: To check if a number is divisible by 13, multiply its unit place digit by 4, then add the product obtained to the number formed by the rest of the digits of the number. If the result is 0 or a multiple of 13, the result is divisible by 13.

Step 1: Multiply the unit place digit by four.

Step 2: Add the product to the left of the units place digit.

Step: 3 If the resulting number is a 0 or a multiple of 13, the number is divisible by 13.

Example: Find out 429 is divisible by 13

Step 1: The one place digit in the number 429 is 9.

Step 2: When we multiply the ones place digit by four, we get 

4×9=36

Step 3: When we add 36 to the remaining digits on the left, we get

42 + 36 = 78.

78 is a multiple of 13, As a result, 429 is divisible by 13.

Divisibility by 13 Rule 3

Rule 3: Step 1: Take a number’s last two digits and form a number.

Step 2: Subtract the product of 4 and the number formed by the remaining digits from the number found in the first step.

Step 3: If the result is 0 or a multiple of 13, we can say the number is divisible by 13.

Example: The last two digits of the number 745 are 45.

Step 1: When we multiply 4 by the remainder of the digits, which is 7, Then we get

Step 2: When we subtract 28 from 45

45-28 = 17, so we get 17.

Step 3: As a result, because 17 is not a multiple of 13, 745 is not divisible by 13.

Divisibility by 13 Rule 4

Rule 4: Multiply the digit in the unit place by 9 and calculate the difference between this product and the remainder of the number to its left. We can say that a number is divisible by 13 if this result is 0 or a multiple of 13.

Example: Following the fourth rule of determining the divisibility of 987 by 13

Step 1: we multiply the last digit (7) by 8 to get 7\times 8 \) which is 56.

Step 2: When we subtract 56 from 98 = (98-56) = 42

Step 3: Then we get 42. Because 35 is not a multiple of 13, 987 is not divisible by 13.

Divisibility Rules for 13 for Large Numbers

Checking if a number is divisible by 13 is quite easy when the number has only two digits or if you know the first few multiples of 13 (like 13, 26, 39, etc.). But what if you get a big number, like a 5-digit number? How do you know if 13 will divide it evenly?

Don't worry! There are a few smart rules (four main ones) that can help you figure it out without actually dividing the number. These rules help you break down the number in a step-by-step way to check if it’s divisible by 13.

Example: Determine whether or not 59371 is divisible by 13 using the divisibility rule of 13.

Solution: Multiply 59371’s last digit by 9, which equals 

9×1=9

The remaining number is 5937.

Subtract 9 from 5937 to get 5928.

5937-9 = 5928

We are unsure whether 5928 is a multiple of 13. Let’s go through it again.

Multiply 5928’s last digit by 9, which equals 

8×9=72

592 – 72 = 520 is the difference between 592 and 72.

We are still unsure whether 520 is a multiple of 13. Let’s go through it again.

Multiply 520 last digit by 9, which equals 

0×9=0

The difference between 52 and 0 is 52, which is a multiple of 13.

As a result, 59371 is divisible by 13.

Divisibility Rules of 13 and 14

The rules for checking if a number can be divided by 13 and 14 are different.

• Rule for 13: Multiply the last digit of the number by 4, then add that to the rest of the number. If the result is divisible by 13, then the original number is also divisible by 13.
   
• Rule for 14: A number must be divisible by both 2 and 7 to be divisible by 14.

Let’s try an example with the number 182:

Divisibility by 13 (for 182):

1. Take the last digit: 2
    
2. Multiply by 4: 2 × 4 = 8
    
3. Add it to the rest of the number (which is 18): 18 + 8 = 26
    
4. Is 26 divisible by 13? Yes → So, 182 is divisible by 13

Divisibility by 14 (for 182):

1. Is 182 divisible by 2? Yes, it's even.
    
2. Check divisibility by 7 using a shortcut:
    
   • Double the last digit (2 × 2 = 4)
      
   • Subtract from the remaining digits: 18 - 4 = 14
      
3. Is 14 divisible by 7? Yes → So, 182 is divisible by 14

Divisibility Rules of 13 and 17 

• For 13, the rule is the same: multiply the last digit by 4 and add it to the rest.
   
• For 17, multiply the last digit by 5 and subtract it from the rest of the number. If the result is divisible by 17, then the number is divisible by 17.

Try with number 204:

Divisibility by 13 (for 204):

1. Last digit = 4, multiply by 4 → 4 × 4 = 16
    
2. Add to rest: 20 + 16 = 36
    
3. Is 36 divisible by 13? No → 204 is not divisible by 13

Divisibility by 17 (for 204):

1. Last digit = 4, multiply by 5 → 4 × 5 = 20
    
2. Subtract from the rest: 20 - 20 = 0
    
3. 0 is divisible by 17 → 204 is divisible by 17

Solved Examples of Divisibility Rules for 13

Example 1: Can you figure out if the smallest four-digit number is divisible by 13?

Solution: 1000 is the smallest four-digit number. Let us use the following rule to determine whether 1000 is divisible by 13 or not.

Subtract the last two digits of a number from the product of 4 and the remainder of the number. If the resulting number is 0 or a multiple of 13, the number is said to be divisible by 13.

The number’s last two digits are 00.

The product of 4 and the remainder of the number (10) is \(10\times 4 \), which equals 40.

Subtraction of 0 from 40 yields 40 – 0, which is 40.

However, 40 is not a multiple of 13. As a result, the smallest four-digit number is not divisible by 13.

Example 2: Is 317 divided by 13?

Solution: Four times the the last digit = \( 4\times 7 \)

Remaining left 31

Now, Add 28 + 31 to get 59

Because 59 is not divisible by 13

317 cannot be divided by 13

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