SOLUTION: Find digits a and b such that the number 34a47b5893 is divisible by 99.

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Question 75181: Find digits a and b such that the number 34a47b5893 is divisible by 99.
Answer by nilan(17) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose 34a47b5893 is divisible by 99
Then obviously it is divisible by 9
If a number is divisible by 9 the sum of all the digits should divisible by 9 (this can be easily proved)
For example let the number abcd divisible by 9 then
10%5E3a%2B10%5E2b%2B10c%2Bd+=+9k where k is positive integer
Then 1000a%2B100b%2B10c%2Bd+=+9k
%28999a%2Ba%29%2B%2899b%2Bb%29%2B%289c%2Bc%29%2Bd=9k
a%2Bb%2Bc%2Bd+=+9k-%28999a%2B99b%2B9c%29
a%2Bb%2Bc%2Bd+=+9%28k-111a%2B11b%2Bc%29=9K
then a+b+d+c is divisible by 9
This can be proved for any number (can be given a general proof. Try it)
Again to the problem
Then the sum of 34a47b5893 is divisible by 9
Then 3+4+a+4+7+b+5+8+9+3 is divisible by 9
Then
43+a+b is divisible by nine
Then a=0 and b=2
Or a=2 and b = 0