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Find the number of pairs of integers (a,b) such that ab = a + b.

 Nov 30, 2019
 #1
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+1

Well, the question didn't say a and b are two distinct numbers right? 

So a possibility would be a=b 

(2,2) works ,(0,0) works 

2*2=2+2 

 

Hence: 

\(a=\frac{b}{b-1}\)

\(b=\frac{a}{a-1}\)

Notice a and b can't equal to 1 

In other words, 

a+b= \(-\frac{b-2ab+a}{(a-1)(b-1)}\)

a-1 can't equal to 1 

b-1 can't equal to 1

 

Leads to the same. 

a=ab-b = b(a-1)

a=b(a-1)

a/b=(a-1) 

ab=a+b

ab-a=b

Notice a and b can equal to 0 and it works as well. 

No other integer solutions other than these. 

https://www.wolframalpha.com/input/?i=ab%3Da%2Bb Graph is a hyberbolic 

 Nov 30, 2019

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