In this section, we’ll explore three methods:
- The last-digit doubling method (best for smaller numbers)
- The chunk (3-digit) alternating sum method (best for 5 digit or larger numbers)
- Mod or remainder alternating sum method
A number is said to be divisible by 7 if it can be divided by 7 without leaving any remainder. Instead of performing long division every time, there is a shortcut rule:
1. For Small Numbers Like 4 Digits or Less – Last Digit Doubling Method
Divisibility Rule for 7:
Take the last digit of the number, double it, and subtract this from the remaining leading part of the number.
- If the result is divisible by 7 (or is 0), then the original number is divisible by 7.
- Repeat the process if needed until you get a smaller number that you can easily check.
Step-by-Step Method
- Separate the last digit of the number.
- Double the last digit.
- Subtract this doubled value from the rest of the digits.
- Check divisibility of the new number by 7.
- Repeat until you reach a small enough number.
Examples of the Divisibility Rule for 7
Example 1: Check if 203 is divisible by 7
- Last digit = 3
- Double it = 6
- Remaining part = 20
- Subtract = 20 – 6 = 14
- 14 ÷ 7 = 2 (perfectly divisible)
Therefore, 203 is divisible by 7.
Example 2: Check if 371 is divisible by 7
- Last digit = 1
- Double it = 2
- Remaining part = 37
- Subtract = 37 – 2 = 35
- 35 ÷ 7 = 5 (perfectly divisible)
So, 371 is divisible by 7.
Example 3: Check if 422 is divisible by 7
- Last digit = 2
- Double it = 4
- Remaining part = 42
- Subtract = 42 – 4 = 38
- 38 ÷ 7 = 5 remainder 3 (not divisible)
Hence, 422 is not divisible by 7.
Shortcut Variations of the Rule
Apart from the standard method, there are two additional approaches to check divisibility by 7:
1. Repeated Subtraction Method
- Keep subtracting 7 from the number until you reach 0 or a small remainder.
- If the final result is 0, the number is divisible.
Example: 49 → 49 – 7 – 7 – 7 … = 0
Yes, Divisible by 7.
2. Direct Division Check (for small numbers)
For smaller numbers under 100, you can directly check with multiplication tables of 7:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98.
If the number is present in this list, it is divisible by 7.
2. For Bigger Numbers like 5 digits or More Chunk (Group) Alternating Sum Method
When numbers reach 5 digits or more, you can use the 3-digit chunk and alternating sum rule. This approach uses modular arithmetic and works faster.
Rule:
- Split the number into 3-digit chunks ( Groups ) starting from the right.
- Assign alternating signs (+, –, +, – …), beginning with the rightmost chunk (Group) as positive.
- Add/subtract the chunks (Groups of 3 digits).
- If the result is divisible by 7, the original number is also divisible by 7.
(Tip: You can reduce each chunk modulo 7 before summing to keep numbers small.)
Example A: 987,654,321
Chunks ( Groups of 3 digits from right) → 321, 654, 987
Alternating sum → 321 – 654 + 987 = 654
Now you can apply last digit doubling method to 654
So we get 65-8 = 57 , which is not divisible by 7
So given number is not divisible by 7.
Example B: 557,664,184
Chunks → 184, 664, 557
Alternating sum → 184 – 664 + 557 = 77
Check → 77 ÷ 7 = 11
Divisible by 7.
3. Mod or Remainder Alternating Sum Method
Example: 666,142,778 (12-digit ready style)
Chunks → 778, 142, 666
Now reduce each chunk mod 7:
- 778 mod 7 = 1 ( Remainder method )
- 142 mod 7 = 2
- 666 mod 7 = 1
Alternating sum → (1 – 2 + 1) = 0
Divisible by 7.
When to Use Which Method?
- Numbers with up to 6 digits → Use the last-digit doubling & subtraction rule (quick and simple).
- Numbers with 9 digits or more → Use the 3-digit chunk alternating sum method (saves time and avoids long calculations).
Both methods give the same result but choosing the right method makes checking divisibility much faster.
Key Takeaways
- Divisibility by 7 is not as straightforward as 2, 3, or 5, but with these rules it becomes easy.
- For small numbers, use the classic last-digit rule.
- For very large numbers, break them into 3-digit chunks and use the alternating sum rule.
- Practicing both methods will make you faster at solving math problems, whether in exams, mental math, or coding tasks.
Why is the Divisibility Rule for 7 Useful?
- Saves time in competitive exams like SSC, Banking, and Olympiads.
- Helps in quick factorization of larger numbers.
- Useful in mental math and aptitude tests.
Quick Practice Questions
- Is 875 divisible by 7?
- Check whether 1197 is divisible by 7.
- Find if 2464 is divisible by 7.
(Try applying the rule step by step!)
When solving math problems, especially in number theory or competitive exams, checking whether a number is divisible by another saves time. While divisibility rules for numbers like 2, 3, 5, and 9 are straightforward, the divisibility rule for 7 often confuses students. We have broke it down step by step with clear methods, explanations, and examples.
The divisibility rule for 7 may seem tricky at first, but with practice, it becomes as simple as the rules for 3 or 9. By using the method of doubling the last digit and subtracting it, you can quickly determine whether a number is divisible by 7 without doing long division.
