Problems need HardWork #9
Number Theory
Level
3
How many integer value of are there such that is a perfect square?
infinite
7
13
0
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Here let 1450 be 2 5 5 29 by factorizIng. Then, let the question be converted as n/(1450-n)=y^2 Or, n(1+y^2)=2 5^2 29 y^2 Here, y^2+1 can only be divisible by 2,5^2,29 and its combinations.. but not by y^2. Now, when y={0,1,2,3,7,12,17}, y^2+1 has the prime factors only of 2,5,29. And we will check for all combination possible that is 12, but for 7 values it satisfy the condition above. Therefore, there are 7 values of y is possible . Thus, 7 values for n is possible only.