Test of Divisibility by 11


According to the test of divisibility rule of 11, we must subtract and then add the digits in an alternating pattern from left to right. If the answer is 0 or 11, then the result is divisible by 11.

Try the problem


Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker\(1A_2\) . What is the missing digit  A of this 3 -digit number?

Source
Competency
Difficulty
Suggested Book

AMC 8

Number System

5 out of 10

Challenges and Thrills in Pre-College Mathematics

Excursion of Mathematics

Knowledge Graph


Test of Divisibility of 11- Knowledge Graph

Use some hints


First hint

Given, eleven members of the Middle School Math Club each paid the same amount for a guest speaker. So the  total amount paid to the guest must be divisible by 11 .Therefore , 1A2 is divisible by 11.

Second Hint

Now remind the test of divisibility by 11.So , 1+2- A  is divisible by 11.

Third Hint

Clearly , 1+2 – A cannot be equal to 11 or any multiple of 11 greater than that as A is a digit and it lies between 0 to 9. Also, if the expression 1+2 – A is equal to a negative multiple of A. A must be 14 or greater, which violated the condition that A is a digit.

Final Step

So, 3-A=0 and 0 is divisible by any number. Hence, A=3.



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