Misreading the question? No problem!

Algebra Level 4

In a math exam, I was instructed to find the product of 2 distinct integers.

However, I misread the question and I found the sum (instead of the product) of these 2 numbers.

If I did my calculation correctly, is it still possible for me to get the correct numerical answer?

Yes, it's possible No, it's impossible

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

If x , y x,y are the two integers, then we require that

x y = x + y ⟹ x y − x − y = 0 ⟹ ( x − 1 ) ( y − 1 ) − 1 = 0 ⟹ ( x − 1 ) ( y − 1 ) = 1 xy = x + y \Longrightarrow xy - x - y = 0 \Longrightarrow (x - 1)(y - 1) - 1 = 0 \Longrightarrow (x - 1)(y - 1) = 1 .

This is only the case if x − 1 = y − 1 = 1 ⟹ x = y = 2 x - 1 = y - 1 = 1 \Longrightarrow x = y = 2 or x − 1 = y − 1 = − 1 ⟹ x = y = 0 x - 1 = y - 1 = -1 \Longrightarrow x = y = 0 .

In either case x , y x,y are not distinct, so if x , y x,y are distinct integers then x y ≠ x + y xy \ne x + y , i.e., the answer is "No, it's impossible".

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