Are you tired of performing long divisions to check if a number is divisible by 17? If so, then this article is for you! Here, we’ll discuss the divisibility rule for 17, a handy trick that can help you determine if a number is divisible by 17 in seconds. From the basics of divisibility rules to the application of the rule for 17, this article will cover everything you need to know.

## What Are Divisibility Rules for 17?

Before we dive into the rule for 17, let’s first understand what divisibility rules are. Divisibility rules are shortcuts or tricks that help us determine if a number is divisible by another number without actually performing the division. These rules are based on the properties of numbers and can save us a lot of time when performing calculations.

## Divisibility Rules for 2, 3, 4, 5, 6, 8, 9, and 10

Before we move on to the rule for 17, let’s quickly review some of the other common divisibility rules.

- Divisibility rule for 2: A number is divisible by 2 if its last digit is even (i.e., 0, 2, 4, 6, or 8).
- Divisibility rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility rule for 4: A number is divisible by 4 if its last two digits are divisible by 4.
- Divisibility rule for 5: A number is divisible by 5 if its last digit is either 0 or 5.
- Divisibility rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility rule for 8: A number is divisible by 8 if its last three digits are divisible by 8.
- Divisibility rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility rule for 10: A number is divisible by 10 if its last digit is 0.

## The Divisibility Rule for 17

Now that we’ve reviewed some of the basic divisibility rules, let’s move on to the rule for 17. To determine if a number is divisible by 17, follow these steps:

- Take the last digit of the number and double it.
- Subtract the result from the rest of the number, excluding the last digit.
- If the result is divisible by 17, then the original number is also divisible by 17.

**Let’s try this rule with an example. Suppose we want to check if 153 is divisible by 17.**

- Double the last digit (3 x 2 = 6).
- Subtract 6 from the rest of the number (15 – 6 = 9).
- Since 9 is not divisible by 17, neither is the original number, 153.

**Let’s try another example. Is 221 divisible by 17?**

- Double the last digit (1 x 2 = 2).
- Subtract 2 from the rest of the number (22 – 2 = 20).
- Since 20 is divisible by 17 (20 ÷ 17 = 1 with a remainder of 3), so is the original number, 221.

## Why Does the Rule Work?

Now that we know the rule for 17, you might be wondering why it works. The rule is based on the fact that 10 is congruent to -5 modulo 17. In other words, if we subtract 5 times any multiple of 17 from a number, the resulting number will be congruent to the original number modulo 17.

Let’s take the example of 221. We can write it as:

221 = (2 x 100) + (2 x 10) + 1

Now, we can apply the rule for 17 by subtracting 2 x 5 = 10 times the last digit 1 from the rest of the number:

221 – 10 x 1 = 211

**We can repeat the process:**

211 = (2 x 100) + (1 x 10) + 1

211 – 10 x 1 = 201

201 = (2 x 100) + 1

Since 201 is divisible by 17, so is 221.

## Advantages of the Divisibility Rule for 17

The divisibility rule for 17 can save a lot of time and effort, especially when dealing with large numbers. Unlike long division, which can be tedious and time-consuming, the rule for 17 allows us to quickly check if a number is divisible by 17 in just a few steps. Moreover, this rule is easy to remember and can be applied without any special tools or equipment.

## Other Applications of the Rule

The rule for 17 can also be used to check if a number is a multiple of 17. To do this, we can simply multiply the last digit of the number by 5 and add it to the rest of the number, excluding the last digit. If the result is a multiple of 17, then the original number is also a multiple of 17.

## FAQs

### Can the rule for 17 be used for any number?

No, the rule for 17 only works for the number 17.

### Can the rule for 17 be used to check if a number is prime?

No, the rule for 17 cannot be used to check if a number is prime.

### Can the rule for 17 be extended to other numbers?

Yes, similar rules can be derived for other numbers, but they may be more complex than the rule for 17.

### Why does the rule for 17 work?

The rule for 17 works because 10 is congruent to -5 modulo 17.

### Are there any other applications of the rule for 17?

Yes, the rule for 17 can also be used to check if a number is a multiple of 17.

## Conclusion

The divisibility rule for 17 is a simple and effective tool for checking if a number is divisible by 17. By following a few easy steps, we can quickly determine if a number is divisible by 17 without performing long divisions. This rule is easy to remember, easy to apply, and can save a lot of time and effort. So next time you need to check if a number is divisible by 17, give this rule a try!