Remainder When Dividing by 6
Last Updated : 07 Nov, 2024
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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