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# help

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

Nov 30, 2019

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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

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