Question

If $a,b$ are natural numbers such that $2013+a^2=b^2$ , the minimum possible value of $ab$ is

• A. $671$
• B. $668$
• C. $658$
• D. $645$

Answer 1

Answer: (C)

$658$

We can write $2013+a^2=b^2$ as

$(b+a)(b-a)=2013$

Factorising $2013$

$(b+a)(b-a)=3\times 11\times 61$

In our case the product of $ab$ is minimum, when the difference between the numbers is minimum. From our factors the minimum product of any two numbers from the given factors is $11\times 3=33$

$\Rightarrow a-b=33$ and $a+b=61$

Solving for $a$ and $b$, we get

$a=47,b=14$

$\therefore ab=658$