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The sum of a positive integer and 100 is a square number. The sum of the same positive integer and 168 is a square number. Find the positive integer.

 

This is a chinese middle school problem

 Jan 30, 2019
edited by Guest  Jan 30, 2019
edited by Guest  Jan 30, 2019
 #1
avatar+101255 
+1

156 + 100  =  256

156 + 168 =   324

 

 

cool cool cool

 Jan 30, 2019
 #2
avatar+101255 
+1

Here is the solution  

 

N + 100 =  P^2

N + 168 = Q^2

 

Subtract the first from the second and we get that

 

Q^2 -  P^2 =  68

(Q + P) (Q - P) = 68

 

The factors that multiply to 68  are

68   1

34   2

17   4

 

But note that we cannot  add/subtract two positive integers and obtain different parities

For example   5 + 2   = 7    and 5 - 2 = 3  ......both  are odd

 

Then....the only possible solution is   Q + P = 34   and Q - P = 2

Adding these we have that 2Q = 36  ⇒   Q = 18

And P = 16

 

So  Q^2 =  324     and P^2 = 256

 

So.....using  N + 100 = P^2 ⇒  N + 100 = 256  ⇒  N = 156

 

And checking

156 + 168  =   324  =  Q^2  = 18^2

 

 

cool cool cool

 Jan 30, 2019
edited by CPhill  Jan 30, 2019
edited by CPhill  Jan 30, 2019

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