Remainder When Divided by 3

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:

• 987
  • Sum of Digits: 9 + 8 + 7 = 24
  • As sum is divisible by 3, thus 987 is divisible by 3 and remainder when divided by 3 is 0.
• 2345
  • Sum of Digits: 2 + 3 + 4 + 5 = 14
  • As sum is not divisible by 3, thus 2345 is also not divisible by 3.
  • Now, remainder when 2345 is divided by 3 is same as when 14 is divided by 14 i.e., 2.

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 10^m. The crucial observation here is that: ( 10^m ) mod 3 = 1 and ( 10^m - 1 ) mod 3 = 0

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

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

Proof:

Let any number be N then,

N = (a_m ⋅ 10^m + a_m−1 ⋅ 10^{m - 1}+ . . . + a₁ ⋅ 10 + a₀) - ( a_m + a_m−1 + . . . + a₁ ⋅+ a) + ( a_m + a_m−1 + . . . + a₁ ⋅+ a)
N = [a_m ⋅ (10^m - 1) + a_m−1 ⋅ (10^{m - 1}- 1) + . . . + a₁ ⋅ (10 - 1) + a₀ ]+ ( a_m + a_m−1 + . . . + a₁ ⋅+ a)

N mod 3 = [ a_m ⋅ (10^m - 1 )+ a_m−1 ⋅ (10^{m - 1}- 1) + . . . + a₁ ⋅ (10 - 1 )] mod 3 + ( a_m + a_m−1 + . . . + a₁ ⋅+ a) mod 3

Since ( 10^m - 1 ) mod 3 = 0 for all positive values of m
[a_m ⋅ (10^m - 1) + a_m−1 ⋅ (10^{m - 1}- 1) + . . . + a₁ ⋅ (10 - 1) ] mod 3 = 0

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

N mod 3 = ( a_m + a_m−1 + . . . + a₁ ⋅+ 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.

Related Articles,

• Divisibility Rule of 3
• Dividend, Divisor, Quotient, and Remainder