Divisibility Rule for 7
How the Divisibility Rule for 7 Works
The divisibility rule for 7 is a quick mental math trick that lets you check if a number is divisible by 7 without performing long division. This is especially useful for large numbers or when you don't have a calculator.
Follow these steps:
- Take the last digit (the unit digit) of the number and multiply it by 2.
- Subtract that product from the remaining part of the number (the number without its last digit).
- If the result is 0 or a multiple of 7, then the original number is divisible by 7.
Note: If the result is still too large to easily determine if it's a multiple of 7, you can repeat the process on the new number until it becomes clear.
Common small multiples of 7 include: \[ 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98 \]
Why does this work? The rule is based on modular arithmetic. A number can be written as \(10a + b\), where \(b\) is the last digit and \(a\) is the remaining part. Since \(10a + b\) is divisible by 7 if and only if \(a - 2b\) is divisible by 7 (because \(10 \equiv 3 \pmod 7\) and \(3 \times 5 \equiv 1 \pmod 7\)), subtracting twice the last digit gives us an easier number to test.
Worked Examples
Example 1: A Standard Application
Is 154 divisible by 7?
- The last digit of 154 is 4.
- The remaining number is 15.
- Subtract twice the last digit from the remaining number: \[ 15 - 2(4) = 15 - 8 = 7 \]
Conclusion: The result, 7, is a multiple of 7. Therefore, 154 is divisible by 7.
(Check: \( 154 \div 7 = 22 \))
Example 2: Using the Rule Twice
Is 903 divisible by 7?
Step 1:
- The last digit of 903 is 3.
- The remaining number is 90.
- Subtract twice the last digit from the remaining number: \[ 90 - 2(3) = 90 - 6 = 84 \]
If you are unsure whether 84 is divisible by 7, repeat the rule on 84.
Step 2:
- The last digit of 84 is 4.
- The remaining number is 8.
- Subtract twice the last digit from the remaining number: \[ 8 - 2(4) = 8 - 8 = 0 \]
Conclusion: Since the final result is 0 (which is divisible by 7), the original number 903 is divisible by 7.
(Check: \( 903 \div 7 = 129 \))
Example 3: Using the Rule Multiple Times on Large Numbers
Is 86,415 divisible by 7?
Step 1:
- Number: 86415 → Remaining: 8641, Last digit: 5
- \[ 8641 - 2(5) = 8641 - 10 = 8631 \]
Step 2: (Apply rule to 8631)
- Number: 8631 → Remaining: 863, Last digit: 1
- \[ 863 - 2(1) = 863 - 2 = 861 \]
Step 3: (Apply rule to 861)
- Number: 861 → Remaining: 86, Last digit: 1
- \[ 86 - 2(1) = 86 - 2 = 84 \]
Conclusion: As seen in Example 2, 84 is divisible by 7. Therefore, 86,415 is divisible by 7.
(Check: \( 86415 \div 7 = 12345 \))
Practice Questions
Determine which of the following numbers are divisible by 7. Click on "View Solution" to check your work.
a) 133
- Apply the rule: \( 13 - 2(3) = 13 - 6 = 7 \)
- Conclusion: The result is 7, which is a multiple of 7. Therefore, 133 is divisible by 7.
b) 178
- Apply the rule: \( 17 - 2(8) = 17 - 16 = 1 \)
- Conclusion: The result is 1, which is NOT a multiple of 7. Therefore, 178 is NOT divisible by 7.
c) 847
- Apply the rule: \( 84 - 2(7) = 84 - 14 = 70 \)
- Conclusion: The result is 70, which is a multiple of 7 (\(7 \times 10\)). Therefore, 847 is divisible by 7.
d) 988
- Step 1: \( 98 - 2(8) = 98 - 16 = 82 \)
- Step 2 (Apply to 82): \( 8 - 2(2) = 8 - 4 = 4 \)
- Conclusion: The final result is 4, which is NOT a multiple of 7. Therefore, 988 is NOT divisible by 7.
e) 10,787
- Step 1: \( 1078 - 2(7) = 1078 - 14 = 1064 \)
- Step 2 (Apply to 1064): \( 106 - 2(4) = 106 - 8 = 98 \)
- (Optional) Step 3 (Apply to 98): \( 9 - 2(8) = 9 - 16 = -7 \)
- Conclusion: The final result is a multiple of 7 (both 98 and -7 are divisible by 7). Therefore, 10,787 is divisible by 7.
Frequently Asked Questions
Why does the divisibility rule for 7 work?
The rule works because \(10 \equiv 3 \pmod 7\), and 2 is the inverse of 3 modulo 7 (since \(3 \times 5 = 15 \equiv 1 \pmod 7\)). When you split a number into \(10a + b\) and compute \(a - 2b\), you are essentially removing multiples of 7 from the original number. If the final result is 0 or a multiple of 7, the original number must also be a multiple of 7.
Can I use division instead of the divisibility rule?
Yes, long division always works. However, the divisibility rule is often faster for large numbers and is specifically useful when you don't have a calculator or need to check quickly without performing the full division.
What if the result is negative?
A negative result is fine. As long as the absolute value of the result is 0 or a multiple of 7 (for example, -7, -14, etc.), the original number is divisible by 7.