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Evaluate probability (Posted on 2018-06-11) Difficulty: 3 of 5
If a and b are positive integers, find the probability that (a^2+b^2)/11 is a positive integer.

No Solution Yet Submitted by Ady TZIDON    
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analytical solution Comment 1 of 1
looking at x^2 mod 11 the possible remainders are


So we need two of these to add to a multiple of 11, since 9+9=18 is the largest sum the only possible multiples are 0 and 11.

The only way to get 0 is with 0+0 and there is no pair of values that add to 11.

Thus the only possible way for a^2+b^2 to be divisible by 11 is if both a and b are also divisible by 11.  Since each of these have a 1/11 chance of being divisible by 11 then the total probability is (1/11)^2 or 1/121

  Posted by Daniel on 2018-06-11 07:52:34
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