Product minus sum

Number Theory Level 4

How many unordered pairs of positive integers $\{a,b\}$ have the property that their product minus their sum is equal to $11$ ?

(The pairs $\{a,b\}$ and $\{b,a\}$ are the same.)

0 3 1 2

1 solution

Aaron Tsai
Apr 25, 2016

We can set up the equation $ab-(a+b)=11$ . Adding $1$ to both sides we get

$ab-a-b+1=12$

$(a-1)(b-1)=12$

The pairs of factors of $12$ are $(1,12)$ , $(2,6)$ , and $(3,4)$ . Therefore, there are $\boxed{3}$ unordered pairs $\left \{ a,b \right \}$ that have the property that their product minus their sum is $11$ .

Nice solution. Note that if $(a,b)$ and $(b,a)$ are the same, then they are known as unordered pairs. I have edited the problem accordingly.

You can use \boxed{3} to create a box around 3.

Pranshu Gaba - 5 years, 1 month ago

Product minus sum

1 solution

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