Lesson Restore the omitted digit in a number in a way that the number is divisible by 11

This Lesson (Restore the omitted digit in a number in a way that the number is divisible by 11) was created by by ikleyn(52906)

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Restore the omitted digit in a number in a way that the number is divisible by 11

Problem 1

In the number 123456_87 one digit was omitted, and you see the blank placeholder in the corresponding position.
Restore the digit in the number in a way that the number is divisible by 11.

Solution

The "divisibility by 11 rule" says: the number is divisible by 11 if and only if the alternate sum of its digits is divisible by 11.
See the lesson Divisibility by 11 rule in this site.

In our case the alternate sum of digits is

1 - 2 + 3 - 4 + 5 - 6 + x - 8 + 7 = -3 + x - 1 = x - 4,

where x stands for the unknown digit.

So, x should be 4 for the sum is divisible by 11, and there is no other opportunity.
Thus there is only one solution: the restored digit is 4 and, hence, the restored number is 123456487.

Solve yourself the next problem.

Problem 2

In the number 12349_92 one digit was omitted, and you see the blank placeholder in the corresponding position.
Restore the digit in the number in a way that the number is divisible by 11.

My other lessons in this site on divisibility numbers are

    - Divisibility by 2 rule
    - Divisibility by 3 rule
    - Divisibility by 4 rule
    - Divisibility by 5 rule
    - Divisibility by 6 rule
    - Divisibility by 9 rule
    - Divisibility by 10 rule
    - Divisibility by 11 rule
    - Restore the omitted digit in a number in a way that the number is divisible by 9
    - Can there be a perfect square ?
    - Math circle level problems on divisibility numbers
    - Math circle level problem on restoring digit in the product of two 16-digit numbers
    - Math circle level problem on finding remainders
    - OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11

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