Remainder When Dividing by 6 (original) (raw)

Last Updated : 23 Jul, 2025

We can use modular arithmetic to find the remainder when any number is divisible by 6.

To determine if a number is divisible by 6 using modular arithmetic, first break the number into its individual digits based on place value. Then, calculate the powers of 10 modulo 6 for each digit’s place value. For example, 100 (mod 6) = 1, 101 (mod 6) = 4, 102 (mod 6) = 4 and so on.

In modulo 6, the powers of 10 is always equivalent to 4 except 100 which is equivalent to 1. This can be seen in the following table:

10n (mod 6)
100	1
101	4
102	4
103	4
104	4
105	4
106	4
107	4

**Note: When dividing by 6, the remainder can only be 0, 1, 2, 3, 4, or 5.

Steps to Find Remainder When Dividing by 6

We can use following simple steps for any number to find the remainder when divided by 6.

**Step 1: Break down the number into its digits and powers of 10.

**Step 2: Calculate the modulo for each term and simplify.

To do this, multiply the sum of all digits except the unit digit by 4, then add the unit digit to this sum. This result is equivalent to taking the modulo 6 of the original number.

Example for Finding Remainder When Dividing by 6

To check if 8274736 is divisible by 6 using modular arithmetic, we need to calculate 8274736 (mod 6). If the result is 0, then the number is divisible by 6.

Step 1: Break the number down

We can express 8274736 in terms of its digits and powers of 10:

8274736 = 8 × 106 + 2 × 105 + 7 × 104 + 4 × 103 + 7 × 102 + 3 × 101 + 6 × 100

Step 2: Calculate the sum modulo 7

Now, we calculate the sum using the modular values of the powers of 10.

8274736 (mod 6) = 8 × 106 + 2 × 105 + 7 × 104 + 4 × 103 + 7 × 102 + 3 × 101 + 6 × 100 (mod 6)

8274736 (mod 6) ≣ 8 × 4 + 2 × 4 + 7 × 4 + 4 × 4 + 7 × 4 + 3 × 4 + 6 × 1 (mod 6) [Used values from the the provided table]

8274736 (mod 6) ≣ 4 × (8 + 2 + 7 + 4 + 7 + 3) + 6 (mod 6)

8274736 (mod 6) ≣ 4 × 31 + 6 (mod 6) = 124 + 6 (mod 6) = 130 (mod 6)

**Step 3: Calculate 130 (mod 6)

Now, divide 130 by 6 and find the remainder:

130 ÷ 6 = 11 remainder 4

So, 130 (mod 6) ≣ 4 (mod 6)

Thus, 8274736 when divided by 6, leaves 4 as remainder.

Example: Find the remainder when 8173 is divided by 6.

Solution:

8173 = 8 × 103 + 1 × 102 + 7 × 101 + 3 × 100

⇒ 8173 (mod 6) ≣ 8 × 4 + 1 × 4 + 7 × 4 + 3 × 1 (mod 6)

⇒ 8173 (mod 6) ≣ (8 + 1 + 7) × 4 + 3 (mod 6) = 16 × 4 + 3 (mod 6) = 67 (mod 6)

⇒ 8173 (mod 6) ≣ 67 (mod 6) = 1 (mod 6) [as 67 = 6 × 11 + 1]

Thus, Remainder when 8173 is divided by 6 is 1.

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