When sums and products coincide
Number Theory
Level
pending
A pair of numbers is considered a "beautiful" pair if its sum equals its product. How many beautiful pairs are there?
Note that the numbers need not to be distinct.
We cannot determine
Infinitely many
One
Two
None
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1 solution
A pair of number a , b can be a "beautiful" pair if it is a solution to the quadratic equation
x 2 − z x + z = 0 ,
where z is any number such that z = a + b . Since there are no restrictions (as to when the pairs have to be rational numbers, integers, etc.) then there are infinitely many pairs that satisfy this equation.