SOLUTION: a^2 - b^2 = 391. Find the number of Integral Solutions? e.g. x + y = 2 has infinite numbers of Integral Solution.

Algebra ->  Equations -> SOLUTION: a^2 - b^2 = 391. Find the number of Integral Solutions? e.g. x + y = 2 has infinite numbers of Integral Solution.      Log On


   

Question 579440: a^2 - b^2 = 391. Find the number of Integral Solutions?
e.g. x + y = 2 has infinite numbers of Integral Solution.
Found 2 solutions by richwmiller, Edwin McCravy:
Answer by richwmiller(17219) About Me  (Show Source):
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor just wrote 4, which is wrong, with
no explanation at all.  I have no idea why he would
do that.

Here is the explanation and answer:

       aČ - bČ = 391

(a - b)(a + b) = 391

The only ways to factor 391 are

        391·1
and      23·17
    
We will first consider a and b to be positive

So we have either the system

system%28a-b=1%2Ca%2Bb=391%29 which has solution a=196, b=195

or

system%28a-b=17%2Ca%2Bb=231%29  which has solution a=20, b=3

Since a and b are squared in the original equation, we may 
choose either sign, so these are the integral solutions:

1. a = 196, b = 195
2. a = 196, b = -195
3. a = -196, b = 195
4. a = -196, b = -195
5. a = 20, b = 3
6. a = 20, b = -3
7. a = -20, b = 3
8. a = -20, b = -3

So there are 8 solutions.

Edwin