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How to Check Divisibility by 13?

Published in Divisibility Rules 2 mins read

To check if a number is divisible by 13, you can use a simple rule involving multiplying the last digit by 4 and adding it to the rest of the number.

The Divisibility Rule for 13

The method to determine if a number is divisible by 13 involves these steps:

  1. Isolate the Units Digit: Identify the digit in the one's place (the rightmost digit).
  2. Multiply by 4: Multiply this units digit by 4.
  3. Add to Remaining Digits: Add the result from step 2 to the number formed by all the digits except the units digit.
  4. Check the Sum: If the new sum is either 0 or divisible by 13, the original number is also divisible by 13.
  5. Repeat if Necessary: If the new sum is still too large, you can repeat steps 1-4 until the number is small enough to determine divisibility by 13.

Example

Let's consider the number 182:

  • Units digit: 2
  • Multiply by 4: 2 * 4 = 8
  • Remaining number: 18
  • Add: 18 + 8 = 26
  • 26 is divisible by 13 (26/13 = 2). Therefore, 182 is divisible by 13.

Another example

Let’s take the number 351:

  • Units digit: 1
  • Multiply by 4: 1 * 4 = 4
  • Remaining number: 35
  • Add: 35 + 4 = 39
  • 39 is divisible by 13 (39/13=3). Therefore, 351 is divisible by 13.

More Detailed Example (Larger Number):

Let's examine a larger number, like 8739:

  1. Units digit: 9
  2. Multiply by 4: 9 * 4 = 36
  3. Remaining number: 873
  4. Add: 873 + 36 = 909

Since 909 is still large, we can repeat the process:

  1. Units digit: 9
  2. Multiply by 4: 9 * 4 = 36
  3. Remaining number: 90
  4. Add: 90 + 36 = 126

Repeat again:

  1. Units digit: 6
  2. Multiply by 4: 6 * 4 = 24
  3. Remaining number: 12
  4. Add: 12 + 24 = 36

36 is not divisible by 13, therefore, 8739 is not divisible by 13.

Summary Table

Step Description Example (Using 8739)
1 Identify the units digit 9
2 Multiply the units digit by 4 9 * 4 = 36
3 Get the remaining digits 873
4 Add the result to the remaining digits 873+36 = 909
5 Repeat if necessary repeat

This method is based on the fact that multiples of 13 can be manipulated in such a way that the divisibility can be determined.

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