A very large multiple of 9 (5's)
If x is an integer where:
5x55x555x5555x55555x...
Has the last 101 digits as 100 5's and 1 x
What can be (or is) the value of x if the number is divisible by 9?
Note: x is part of the number. For example, the x at the end of the number means x ones.
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1 solution
An integer is a multiple of 9 if and only is the sum of it's digits is a multiple of 9. We try to find the value(s) of x by adding all the digits.
First we add the fives. (1+2+3+...+100)=5050x5=25250
Next we group the x's so that we get an expression:
25250+100x
That value can be written as:
25"2+x"50 (2x10000+5x1000+2x100+100x+5x10+0)
We add the digits of that number to get a final expression:
14+x
Since the value of 14+x has to be a multiple of 9, we find that the only possible value of x can be is 4, so 4 is the answer.