How to Know a Number is Divisible by 11?
To determine if a number is divisible by 11, calculate the alternating sum of its digits. If this alternating sum is divisible by 11, then the original number is also divisible by 11.
The method involves:
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Alternating Sum: Starting from the rightmost digit, alternately add and subtract the digits of the number. For example, in the number 8361, you would calculate: 1 - 6 + 3 - 8 = -10.
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Divisibility Check: If the resulting sum (or difference) is divisible by 11 (meaning it's 0, 11, -11, 22, -22, etc.), then the original number is divisible by 11. In the example above, -10 is not divisible by 11, therefore 8361 is not divisible by 11.
Let's examine some examples:
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Example 1: 9352
- Alternating sum: 2 - 5 + 3 - 9 = -9. -9 is not divisible by 11.
- Therefore, 9352 is not divisible by 11.
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Example 2: 23671
- Alternating sum: 1 - 7 + 6 - 3 + 2 = -1. -1 is not divisible by 11.
- Therefore, 23671 is not divisible by 11.
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Example 3: 121
- Alternating sum: 1 - 2 + 1 = 0. 0 is divisible by 11.
- Therefore, 121 is divisible by 11.
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Example 4: 71284
- Alternating sum: 4 - 8 + 2 - 1 + 7 = 4. 4 is not divisible by 11.
- Therefore 71284 is not divisible by 11.
Why this works (Brief Explanation)
This method works due to the properties of the base-10 number system and how powers of 10 relate to 11. A rigorous mathematical proof would involve modular arithmetic, but the essence is that the alternating sum captures the relevant information regarding divisibility by 11. References such as Cuemath, Byjus, and others explain this concept in more detail.
