Divisibility rules are shortcuts that help you determine if a number is divisible by a certain integer without actually performing the division. These rules can come in handy, especially when dealing with large numbers. One such rule is the divisibility rule for 19. In this article, we will explore this rule and learn how to apply it.

## What is the Divisibility Rule for 19?

The divisibility rule for 19 states that a number is divisible by 19 if and only if the difference between the number formed by its last two digits and four times the remaining digits is divisible by 19.

For example, let’s take the number 1156. The last two digits of this number are 56, and the remaining digits are 11. According to the rule, we need to subtract four times the remaining digits (11 x 4 = 44) from the number formed by the last two digits (56).

**56 – 44 = 12**

Since 12 is not divisible by 19, 1156 is not divisible by 19.

## Why Does the Divisibility Rule for 19 Work?

The divisibility rule for 19 is based on the fact that 100 is equivalent to -5 modulo 19. This means that if we subtract a multiple of 19 from 100, we get a number that is congruent to -5 modulo 19. For example, 100 – 19 = 81, which is congruent to -5 modulo 19.

Let’s take the number 1156 again and see how this works. We can write 1156 as:

1156 = 1100 + 56

Now, 1100 is a multiple of 100, so we can write it as:

1100 = 11 x 100

Substituting this value in the original equation, we get:

1156 = 11 x 100 + 56

Now, we know that 100 is equivalent to -5 modulo 19. So, we can write:

1156 = 11 x (-5) + 56 (mod 19)

Simplifying this expression, we get:

1156 = -55 + 56 (mod 19)

1156 = 1 (mod 19)

Since 1 is not divisible by 19, 1156 is not divisible by 19.

## Applying the Divisibility Rule for 19

Let’s take a few more examples and see how to apply the divisibility rule for 19.

### Example 1: Is 1939 divisible by 19?

The last two digits of 1939 are 39, and the remaining digits are 19.

39 – 4 x 19 = -17

Since -17 is equivalent to 2 modulo 19, which is not divisible by 19, 1939 is not divisible by 19.

### Example 2: Is 56215 divisible by 19?

The last two digits of 56215 are 15, and the remaining digits are 562.

15 – 4 x 562 = -2243

Since -2243 is equivalent to 5 modulo 19, which is not divisible by 19, 56215 is not divisible by 19.

### Example 3: Is 13943 divisible by 19?

The last two digits of 13943 are 43, and the remaining digits are 139.

43 – 4 x 139 = -549

Since -549 is equivalent to 4 modulo 19, which is divisible by 19, 13943 is divisible by 19

## Advantages of Using the Divisibility Rule for 19

Using the divisibility rule for 19 can save you a lot of time and effort, especially when dealing with large numbers. It can also be useful in various mathematical problems, such as finding factors and multiples of a number.

## Limitations of the Divisibility Rule for 19

The divisibility rule for 19 only works for checking the divisibility by 19. It cannot be used to check divisibility by other numbers. Also, the rule may not work for certain numbers that do not follow the pattern, so it is always a good idea to double-check by performing the division if you are unsure.

## Other Divisibility Rules

There are other divisibility rules for different numbers, such as the rule for 2 (even numbers), the rule for 3 (sum of digits is divisible by 3), the rule for 4 (last two digits are divisible by 4), and the rule for 5 (ends with 0 or 5). Learning these rules can make mathematical calculations much easier and faster.

## FAQs

**What is the difference between the number formed by the last two digits and four times the remaining digits in the divisibility rule for 19?**

The difference between the number formed by the last two digits and four times the remaining digits is used to determine whether a number is divisible by 19. If the difference is divisible by 19, then the original number is also divisible by 19.

**Can the divisibility rule for 19 be used to check divisibility by other numbers?**

No, the divisibility rule for 19 only works for checking the divisibility by 19. There are other divisibility rules for different numbers.

**What are some other divisibility rules?**

Some other divisibility rules include the rule for 2 (even numbers), the rule for 3 (sum of digits is divisible by 3), the rule for 4 (last two digits are divisible by 4), and the rule for 5 (ends with 0 or 5).

**Can the divisibility rule for 19 be used for any number?**

The divisibility rule for 19 may not work for certain numbers that do not follow the pattern, so it is always a good idea to double-check by performing the division if you are unsure.

**Why are divisibility rules important?**

Divisibility rules can make mathematical calculations much easier and faster, especially when dealing with large numbers. They are also useful in various mathematical problems, such as finding factors and multiples of a number.

## Conclusion

The divisibility rule for 19 is a useful shortcut for checking whether a number is divisible by 19. It is based on the difference between the number formed by its last two digits and four times the remaining digits. While it can save you time and effort, it may not work for all numbers, so it is always a good idea to double-check by performing the division.