Remainder Fact For Number 9

Remainder Fact For Number 9

The Remainder Factor for Number 9 is a concept that helps find the remainder when a number is divided by 9. The remainder fact for number 9 helps determine the remainder when a number is divided by 9 using the sum of its digits. This method simplifies the division process and provides an intuitive approach to finding remainders.

Essentially, the remainder when a number is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.

Below are interesting facts and examples to prove the Remainder fact of number 9.

Proof with Example

For any number say 5463, we can re-write it as 5 x 10³ + 4 x 10² + 6 x 10¹ + 3

Any number can be expressed in terms of its digits. For example, the number n = 5463, can be expressed as 5 x 10³ + 4 x 10² + 6 x 10¹ + 3

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 9 = 1 and ( 10^m - 1 ) mod 9 = 0

This means 10^m modulo 9 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 9 = [ a_m ⋅ (10^m - 1 )+ a_m−1 ⋅ (10^{m - 1}- 1) + . . . + a₁ ⋅ (10 - 1 )] mod 9 + ( a_m + a_m−1 + . . . + a₁ ⋅+ a) mod 9

Since ( 10^m - 1 ) mod 9 = 0 for all positive values of m
[a_m ⋅ (10^m - 1) + a_m−1 ⋅ (10^{m - 1}- 1) + . . . + a₁ ⋅ (10 - 1) ] mod 9 = 0

N mod 9 = 0 + ( a_m + a_m−1 + . . . + a₁ ⋅+ a) mod 9

N mod 9 = ( a_m + a_m−1 + . . . + a₁ ⋅+ a) mod 9

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.

So the remainder of a number when divided by 9 is same as the remainder obtained when its digit sum is divided by 9.

thus, 5463 mod 9 = (5 + 4 + 6 + 3) mod 9

For example:

• For the number 5,432
  • Sum of digits : 5 + 4 + 3 + 2 = 14
  • Calculation: 14 mod 9 = 5.
  • Thus, the remainder when 5,432 is divided by 9 is also 5.

• For the Number: 1,234
  • Sum of digits : 1 + 2 + 3 + 4 = 10
  • Calculation: 10 mod 9 = 1.
  • Thus, the remainder when 1,234 is divided by 9 is also 1.

Interesting Facts

• The digital root of a number (repeatedly summing the digits of a number until a single digit is obtained) directly tells whether a number is divisible by 9. If the digital root is 9, the number is divisible by 9. Otherwise, the digital root is the remainder when the number is divided by 9.

Example: Consider 258:
2 + 5 + 8 = 15
1 + 5 = 6.

The digital root is 6, so the remainder when 258 is divided by 9 is 6.

• This rule is basically the divisibility rule of 9, which states that if the sum of digits of a number is divisible by 9, then that number is also divisible by 9. In case the number is not a multiple of 9, this rule is used to find the remainder.

Solved Examples:

Example 1: Find the remainder when 1347 is divided by 9.

Sum of digits: 1 + 3 + 4 + 7 = 15

Divide 15 by 9: Remainder is 6.

So, the remainder of 1347 ÷ 9 is 6.

Example 2: Find the remainder when 986 is divided by 9.

Sum of digits: 9 + 8 + 6 = 23

Divide 23 by 9: Remainder is 5.

So, the remainder of 986 ÷ 9 is 5.

Example 3: Find the remainder when 517 is divided by 9.

Sum of digits: 5 + 1 + 7 = 13

Divide 13 by 9: Remainder is 4.

So, the remainder of 517 ÷ 9 is 4.

Example 4: Find the remainder when 289 is divided by 9.

Sum of digits: 2 + 8 + 9 = 19

Divide 19 by 9: Remainder is 1.

So, the remainder of 289 ÷ 9 is 1.

Example 5: Find the remainder when 999 is divided by 9.

Sum of digits: 9 + 9 + 9 = 27

Divide 27 by 9: Remainder is 0.

So, the remainder of 999 ÷ 9 is 0.

Practice Problems:

1. Find the remainder when 4321 is divided by 9.
2. Find the remainder when 5678 is divided by 9.
3. Find the remainder when 729 is divided by 9.
4. Find the remainder when 3456 is divided by 9.
5. Find the remainder when 987 is divided by 9.
6. Find the remainder when 246 is divided by 9.
7. Find the remainder when 8888 is divided by 9.
8. Find the remainder when 1357 is divided by 9.
9. Find the remainder when 2468 is divided by 9.
10. Find the remainder when 12345 is divided by 9.

Summary:

The Remainder Fact for 9 provides a quick and effective way to determine the remainder when dividing a number by 9 by simply summing its digits. This principle underlies the divisibility rule for 9 and can be extended to solve related problems with ease. If the sum of the digits is divisible by 9, the number is divisible by 9. Otherwise, the remainder of the sum divided by 9 is the remainder when the number itself is divided by 9.

Related Articles:

• Divisibility Rule of 9
• Dividend, Divisor, Quotient and Remainder