As kids, we all learned to divide numbers by 2, 5, and 10 with ease. But when it comes to dividing numbers by 7, we often find ourselves struggling. Well, worry not, because there is a simple trick, known as the divisibility rule for 7, that can make the process much easier. In this article, we’ll delve into the details of this rule and explain how it works.
What is Divisibility?
Before we get into the specifics of the rule, let’s first understand what divisibility is. Divisibility is the property of a number that allows it to be divided by another number without leaving a remainder. For instance, 12 is divisible by 2, 3, 4, and 6, but not by 5 or 7.
The Divisibility Rule for 7
The divisibility rule for 7 is a simple trick that allows us to determine whether a number is divisible by 7 or not. To apply this rule, we need to follow these steps:
Step 1: Double the Last Digit
Take the last digit of the number and double it. For example, if the number is 378, the last digit is 8, so we double it to get 16.
Step 2: Subtract the Result from the Rest of the Number
Now, we subtract the result from step 1 from the rest of the number (excluding the last digit). For example, if the number is 378, we subtract 16 from 37 to get 21.
Step 3: Repeat Until the Result is Either 0 or Divisible by 7
If the result is not divisible by 7, we repeat steps 1 and 2 until we get a result that is either 0 or divisible by 7. For instance, if the number is 378, we would repeat steps 1 and 2 with the result of step 2 (21) to get 2. We can stop here because 2 is not divisible by 7.
Let’s take another example. If the number is 259, we double the last digit (18), subtract it from the rest of the number (25-18=7), and since 7 is divisible by 7, the original number (259) is also divisible by 7.
Why Does the Rule Work?
You may wonder why this rule works. Well, the answer lies in the mathematical concept of modular arithmetic. In modular arithmetic, we work with remainders. For instance, when we divide 17 by 5, the remainder is 2, which is denoted by 17 mod 5 = 2.
Using this concept, we can express any number in the form of a sum of multiples of 10. For example, 378 can be expressed as 3×100 + 7×10 + 8. If we apply the rule for each digit separately, we get:
- For 8: 2×8 = 16, 37-16=21, 2×1=2
- For 7: 2×7=14, 3-14=-11 (or 9 mod 7), 2×9=18, 1-18=-17 (or 3 mod 7)
As you can see, in both cases, we end up with a number that is either 0 or divisible by 7.
Using the Rule for Large Numbers
The divisibility rule for 7 is particularly useful when dealing with large numbers. However, it can become tedious to apply the rule repeatedly. To simplify the process, we can use the fact that we can add or subtract multiples of 7 without changing the divisibility of the number. For example, let’s say we want to determine whether 105 is divisible by 7. We can use the following steps:
- Double the last digit: 5 x 2 = 10
- Subtract the result from the rest of the number: 10 – 10 = 0
Since the result is 0, we know that 105 is divisible by 7. However, if we have a larger number like 968, we can use the fact that 7 x 100 = 700 to simplify the process:
- Double the last digit: 8 x 2 = 16
- Subtract the result from the rest of the number: 96 – 16 = 80
- Subtract a multiple of 7 to simplify the number: 80 – 70 = 10
- Double the last digit: 0 x 2 = 0
Since the result is 0, we know that 968 is divisible by 7.
Practice Problems
Now that we’ve explained the divisibility rule for 7, it’s time to practice it with some problems. Here are a few numbers for you to try:
- 259
- 476
- 539
- 721
Remember to follow the steps we outlined above and repeat them until you get a result that is either 0 or divisible by 7.
Conclusion
In conclusion, the divisibility rule for 7 is a simple trick that can help us determine whether a number is divisible by 7 or not. By doubling the last digit and subtracting it from the rest of the number, we can repeat the process until we get a result that is either 0 or divisible by 7. This rule is particularly useful when dealing with large numbers and can simplify the process of division. So, next time you’re faced with a number that you need to divide by 7, remember the divisibility rule and make your life a little easier.
FAQs
What is the divisibility rule for 7?
The divisibility rule for 7 is a trick that allows us to determine whether a number is divisible by 7 or not by doubling the last digit and subtracting it from the rest of the number.
Why does the divisibility rule for 7 work?
The rule works because of the mathematical concept of modular arithmetic, where we work with remainders.
Can the divisibility rule for 7 be applied to any number?
No, the rule is specific to the number 7 and cannot be applied to other numbers.
Is the divisibility rule for 7 always accurate?
Yes, the rule is always accurate as long as it is applied correctly.
Is there a shortcut for applying the divisibility rule for 7 to large numbers?
Yes, by adding or subtracting multiples of 7, we can simplify the process of applying the rule to large numbers.