Lesson Restore the omitted digit in a number in a way that the number is divisible by 9

This Lesson (Restore the omitted digit in a number in a way that the number is divisible by 9) was created by by ikleyn(52906)

: View Source, Show
About ikleyn:

Restore the omitted digit in a number in a way that the number is divisible by 9

Problem 1

In the number 12345_67 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 9.

Solution

The "divisibility by 9 rule" says: the number is divisible by 9 if and only if the sum of its digits is divisible by 9.
See the lesson Divisibility by 9 rule in this site.

In our case the sum of digits is

1 + 2 + 3 + 4 + 5 + x + 6 + 7 = 28 + x,

where x stands for the unknown digit.

So, x should be 8 for the sum is divisible by 9, and there is no other opportunity.
Thus there is only one solution: the restored digit is 8 and, hence, the restored number is 12345867.

Solve yourself the next problem.

Problem 2

In the number 12345_678 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 9.
Is the solution unique? How many solutions are there?

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 11
    - 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

This lesson has been accessed 2685 times.