Remainder When Dividing by 8

Last Updated : 23 Jul, 2025

The Remainder Rule for 8 is a straightforward technique to find the remainder when dividing a number by 8.

Essentially, the remainder when a number is divided by 8 can be determined by examining the last three digits of that number.

While dividing any number by 8, the only possible remainders are 0, 1, 2, 3, 4, 5, 6, and 7.

For example: Check remainder of 12345678 when divided by 8.

  1. Step 1: Take the last three digits 678.
  2. Step 2: Now, divide 678 by 8: 678 ÷ 8 = 84 with a remainder of 6.

Thus, the remainder when 12345678 is divided by 8 is also 6.

Some other examples include:

  • For the number 5,432
    • Last three digits: 432
    • Calculation: 432 ÷ 8 = 54 with a remainder of 0.
    • Thus, the remainder when 5,432 is divided by 8 is also 0.
  • For the Number: 1,234
    • Last three digits: 234
    • Calculation: 234 ÷ 8 = 29 with a remainder of 2.
    • Thus, the remainder when 1,234 is divided by 8: 2
  • For the Number: 9,999
    • Last three digits: 999
    • Calculation: 999 ÷ 8 = 124 with a remainder of 7.
    • Thus, the remainder when 9,999 is divided by 8: 7
  • For the Number: 24,789
    • Last three digits: 789
    • Calculation: 789 ÷ 8 = 98 with a remainder of 5.
    • Thus, the remainder when 24,789 is divided by 8: 5

Proof of Remainder Rule When Dividing by 8

Let any number be N, then N can be written as 1000Q + d where d represents the last three digits of the number ( Ex: 8542 = 8 × 1000 + 542 here Q = 8 and d = 542 )

N = 1000Q + d
⇒ N mod 8 = ( 1000Q ) mod 8 + d mod 8
⇒ N mod 8 = ( 8 × 125Q ) mod 8 + d mod 8

N mod 8 = 0 + d mod 8

Thus we can say that to find the remainder when dividing by 8 is equal to remainder obtained by dividing the number formed by the last 3 digits of the number.

Remainder for Binary Numbers

When you convert a number to binary, the last three digits (or bits) give you a quick way to find out what the remainder will be when you divide that number by 8.

Example:

Take the number 125. Its binary form is 1111101. The last three bits are 101.

If we convert 101 back to decimal, 101 = 22 + 21 + 20 = 5.

Now, if we divide 125 by 8, the remainder is also 5 (because 125 ÷ 8 = 15 (remainder 5).

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