Remainder When Dividing by 7 (original) (raw)

Last Updated : 23 Jul, 2025

To determine if a number is divisible by 7 using modular arithmetic, first break the number into its individual digits based on place value. Then, calculate the powers of 10 modulo 7 for each digit’s place value. For example, 100 (mod 7) ≣ 1, 101 (mod 7) ≣ 3, and so on.

Multiply each digit by the corresponding power of 10 modulo 7, and sum these values. Finally, find the result of the sum modulo 7. If the final result is 0, the number is divisible by 7; if not, it is not divisible by 7.

In modulo 7, the powers of 10 repeat a pattern of {3, 2, 6, 4, 5, 1}, meaning that every sixth power of 10 is equivalent to one of the values in this pattern under modulo 7.

Power of 10	10n (mod 7)
100	1
101	3
102	2
103	6
104	4
105	5
106	1
107	3
108	2
109	6
1010	4
1011	5

**Note: We can use {1, 3, 2, -1, -3, -2} as series as well as -1 ≣ 6 (mod 7), -2 ≣ 5 (mod 7), and -3 ≣ 4 (mod 7).

Example for Finding Remainder When Dividing by 7

To check if 928386 is divisible by 7 using modular arithmetic, we need to calculate 928386 (mod 7). If the result is 0, then the number is divisible by 7.

Step 1: Break the number down

We can break the number 928386 into its individual digits and use the property of modular arithmetic for each place value.

928386 = 9 × 105 + 2 × 104 + 8 × 103 + 3 × 102 + 8 × 101 + 6 × 100

Step 2: Calculate the sum modulo 7

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

928386 (mod 7) = 9 × 105 + 2 × 104 + 8 × 103 + 3 × 102 + 8 × 101 + 6 × 100 (mod 7)

928386 (mod 7) ≣ 9 × 5 + 2 × 4 + 8 × 6 + 3 × 2 + 8 × 3 + 6 × 1 (mod 7) [Used values from the the provided table]

928386 (mod 7) ≣ 45 + 8 + 48 + 6 + 24 + 6 (mod 7) = 137 (mod 7)

Step 3: Calculate 137 ((mod 7))

Now, divide 137 by 7 and find the remainder:

137 ÷ 7 = 19 remainder 4

So, 137 (mod 7) ≣ 4 (mod 7)

Thus, 928386 when divided by 7, leaves 4 as remainder.

**Example: Find remainder when 4389274 is divided by 7.

**Start with result = 0.

**Digits: 4, 7, 2, 9, 8, 3, 4 (from right to left).

**Series: 1, 3, 2, -1, -3, -2 (repeats every 6 steps).

Now, multiply each number with series respectively:

**4 × **1 = 4,

**7 × **3 = 21,

**2 × **2 = 4,

**9 × **-1 = -9,

**8 × **-3 = -24,

**3 × **-2 = -6,

**4 × **1 = 4,

Thus, **4389274 (mod 7) ≣ 4 + 21 + 4 - 9 - 24 - 6 + 4 (mod 7) ≣ -6 (mod 7) ≣ 1 (mod 7)

So, the remainder when **4389274 is divided by 7 is **1.

Remainder When Dividing by 7 for Larger Numbers

To find remainder when dividing by 7 for larger number we can use the following steps:

Break the number into groups of three and calculate modulo 7 for each.
Take the alternating sum of the remainders.
Find the modulo 7 of the alternating sum to get the final result.

**Example: Find remainder when 7864271624889 is divided by 7.

**Solution:

Split **7864271624889 into group of three i.e., 7,864,271,624,889.

Group	7	846	271	624	889
Remainder (mod 7)	0	3	5	1	0

Alternating Sum = 0 - 3 + 5 - 1 + 0 = 1 (mod 7)

Thus, remainder when 7864271624889 is divided by 7 is 1.

**Read More,