Divisibility rules are a valuable tool in mathematics, allowing us to quickly determine if a number is divisible by another number without performing long division. Among these rules, the rule for divisibility by 97 might seem a bit tricky at first, but it's actually quite straightforward once you understand the logic behind it. In this article, we will explore the intriguing method of checking divisibility by 97, explaining the reasoning and providing practical examples to solidify your understanding.
The Intriguing Logic Behind Divisibility by 97
The divisibility rule for 97 hinges on a clever trick that involves subtracting a multiple of 97 from the original number. The key principle is that if the difference is divisible by 97, then the original number is also divisible by 97. This rule is based on the fact that if a number is divisible by 97, then subtracting a multiple of 97 will not change its divisibility.
The Steps to Check Divisibility by 97
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Double the last digit of the number: Take the last digit of your original number and multiply it by 2.
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Subtract the doubled digit from the remaining part of the number: Remove the last digit from the original number and subtract the doubled digit from this remaining part.
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Repeat steps 1 & 2 until you get a number you recognize: Continue repeating steps 1 and 2 on the resulting number until you reach a number you know is divisible by 97 or a number you know is not divisible by 97.
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Determine divisibility: If the final number is divisible by 97, then the original number is also divisible by 97. If the final number is not divisible by 97, then the original number is not divisible by 97.
Illustrative Examples to Clarify the Process
Let's consider a few examples to illustrate how the divisibility rule for 97 works in practice:
Example 1: Is the number 123456 divisible by 97?
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Double the last digit (6): 6 x 2 = 12
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Subtract the doubled digit from the remaining part (12345): 12345 - 12 = 12333
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Repeat steps 1 & 2:
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Double the last digit (3): 3 x 2 = 6
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Subtract the doubled digit from the remaining part (1233): 1233 - 6 = 1227
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Double the last digit (7): 7 x 2 = 14
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Subtract the doubled digit from the remaining part (122): 122 - 14 = 108
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Repeat steps 1 & 2:
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Double the last digit (8): 8 x 2 = 16
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Subtract the doubled digit from the remaining part (10): 10 - 16 = -6
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Double the last digit (6): 6 x 2 = 12
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Subtract the doubled digit from the remaining part (-0): -0 - 12 = -12
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Determine divisibility: Since -12 is not divisible by 97, the original number 123456 is not divisible by 97.
Example 2: Is the number 588235 divisible by 97?
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Double the last digit (5): 5 x 2 = 10
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Subtract the doubled digit from the remaining part (58823): 58823 - 10 = 58813
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Repeat steps 1 & 2:
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Double the last digit (3): 3 x 2 = 6
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Subtract the doubled digit from the remaining part (5881): 5881 - 6 = 5875
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Double the last digit (5): 5 x 2 = 10
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Subtract the doubled digit from the remaining part (587): 587 - 10 = 577
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Double the last digit (7): 7 x 2 = 14
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Subtract the doubled digit from the remaining part (57): 57 - 14 = 43
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Determine divisibility: Since 43 is not divisible by 97, the original number 588235 is not divisible by 97.
A Word on Simplifying the Process
While repeating the steps might seem repetitive, keep in mind that you can often stop the process early if you encounter a number you know is divisible by 97. For instance, if you end up with a number like 97, you can immediately conclude that the original number is divisible by 97 without needing to continue further. Additionally, if you happen to arrive at a number you know is not divisible by 97, like 10 or 50, you can stop the process and determine that the original number is not divisible by 97.
Conclusion
Mastering the divisibility rule for 97 can be a handy skill for various mathematical tasks, from simplifying calculations to identifying prime numbers. While the process might appear slightly complex at first glance, it's quite simple and can be quickly memorized with practice. By understanding the logic behind this intriguing rule and following the outlined steps, you can easily assess the divisibility of any number by 97 without resorting to lengthy division. So, next time you encounter a large number and need to determine its divisibility by 97, remember the steps and apply the rule confidently!
