Divisibility by 11: The Easy Trick

Divisibility by 11: The Easy Trick

Divisibility rules are handy shortcuts to determine if a number is divisible by another number without actually performing the division. For the number 11, there's a neat trick that makes checking divisibility a breeze. Let's explore it together.

The Trick

The trick involves alternating the signs of the digits in the number and adding them together. If the result is divisible by 11, then the original number is also divisible by 11.

Example 1

Let's take the number 121.

1. Alternating Signs: 1 - 2 + 1
2. Adding the Digits: 1 - 2 + 1 = 0
3. Divisibility Check: 0 is divisible by 11.

Therefore, 121 is divisible by 11.

Example 2

Let's try another example with the number 3432.

1. Alternating Signs: 3 - 4 + 3 - 2
2. Adding the Digits: 3 - 4 + 3 - 2 = 0
3. Divisibility Check: 0 is divisible by 11.

Therefore, 3432 is divisible by 11.

Why Does This Work?

The divisibility rule for 11 stems from the fact that powers of 10 alternate between leaving a remainder of 1 and 10 when divided by 11. Here's a breakdown:

• 10⁰ = 1 (remainder 1 when divided by 11)
• 10¹ = 10 (remainder 10 when divided by 11)
• 10² = 100 (remainder 1 when divided by 11)
• 10³ = 1000 (remainder 10 when divided by 11)

When you alternate the signs of the digits in a number, you're essentially subtracting the values of the digits in the odd places and adding the values of the digits in the even places. This manipulation aligns with the alternating remainders of the powers of 10 when divided by 11, making the divisibility check possible.

When the Sum is Not Divisible by 11

If the sum of the alternating digits is not divisible by 11, then the original number is also not divisible by 11.

Example

Let's take the number 234.

1. Alternating Signs: 2 - 3 + 4
2. Adding the Digits: 2 - 3 + 4 = 3
3. Divisibility Check: 3 is not divisible by 11.

Therefore, 234 is not divisible by 11.

Practice Makes Perfect

The best way to master this divisibility trick is to practice. Try applying it to different numbers. You'll be surprised how quickly you can determine if a number is divisible by 11.

Additional Tips

• If the sum of the alternating digits is a multiple of 11, then the original number is divisible by 11.
• If the sum of the alternating digits is a negative number, you can take its absolute value and check if it's divisible by 11.

With this simple trick, you can confidently determine the divisibility of a number by 11. It's a helpful tool to have in your mathematical arsenal!