Find all positive integers such that

Question

n² + 96 is a perfect square.

Answer 1

We know that perfect squares are values that are positive and rational.  Some examples of perfect squares are 1 , 4 , 9 , 16 , 25.

 

 

So we can set some equations using these values.

 

n² + 96 = 1

n² = -95

 

 

No number squared gives us a negative number.  So we need to pick square values greater than 96, based on the algebra you saw here.  Numbers we can pick are 100 , 121 , 144 , 169 , 196.  When we solve for n by using these values and above into equation, we will see a pattern for n.

 

 

n² + 96 = 100

n² = 4

n = 2

 

 

n² + 96 = 121

n² = 25

n = 5

 

 

n² + 96 = 144

n² = 48

 

48 is not a perfect square.

 

 

 

n² + 96 = 169

n² = 73

 

73 is not a perfect square.

 

 

n² + 96 = 196

n² = 100

n = 10

 

 

If we continue to use perfect square values in ascending order, we will see that when we solve for n, n will be an irrational number.  So

 

n = 2     ,     n = 5   ,       and          n = 10