Remainder When Divided by 3 (original) (raw)

Last Updated : 23 Jul, 2025

The Remainder Rule for 3 is a simple way to find the remainder when a number is divided by 3.

**Essentially, the remainder when a number is divided by 3 is the same as the remainder of the sum of its digits when divided by 3.

While dividing any number by 3 the only possible remainders are 0, 1, 2.

**For Example:

Proof for Remainder When Divided by 3

Any number can be expressed in terms of its digits. For example, the number n = 253 can be expressed as: n = 2 × 100 + 5 × 10 + 3 × 1

In base 10, each place value (like 100, 10, and 1) can be expressed as 10m. The crucial observation here is that: ****( 10** m ) mod 3 = 1 and ****( 10** m **- 1 ) mod 3 = 0

This means 10m modulo 3 is equal to 1 for any positive integer m.

When considering the number n = amam−1 . . . a1a0 (where ai​ are the digits of the number):

**Proof:

**Let any number be N then,

**N = (a m **⋅ 10 m **+ a m−1 **⋅ 10 m - 1 + . . . + a 1 **⋅ 10 + a 0 ) - ( am + am−1 + . . . + a1 ⋅+ a) + ( am + am−1 + . . . + a1 ⋅+ a)
N = [am ⋅ (10m - 1) + am−1 ⋅ (10m - 1- 1) + . . . + a1 ⋅ (10 - 1) + a0 ]+ ( am + am−1 + . . . + a1 ⋅+ a)

N **mod 3 = [ a m **⋅ (10 m **- 1 )+ a m−1 **⋅ (10 m - 1 - 1) + . . . + a 1 **⋅ (10 - 1 )] mod 3 + ( a m **+ a m−1 **+ . . . + a 1 **⋅+ a) mod 3

**Since ( 10 m **- 1 ) mod 3 = 0 for all positive values of m
[am ⋅ (10m - 1) + am−1 ⋅ (10m - 1- 1) + . . . + a1 ⋅ (10 - 1) ] mod 3 = 0

N mod 3 = 0 + ( am + am−1 + . . . + a1 ⋅+ a) mod 3

**N mod 3 = ( a m **+ a m−1 **+ . . . + a 1 **⋅+ a) mod 3

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

This means that n mod 3 is equivalent to the sum of its digits S modulo 3: n mod 3 = S mod 3

**Note: This rule works for number to find the remainder when divided by 3.

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