Question

If $a$ and $b$ are positive integers such that $a^2-b^2$ is a prime number, then $a^2-b^2$ is

• A. $a+b$
• B. $a-b$
• C. $ab$
• D. 1

Answer 1

Answer: (A)

$a$ and $b$ are positive integers and $a^2-b^2$ is a prime number.
Since, $a^2-b^2=(a+b)(a-b)$ product of two numbers.
$\therefore$ Either $a+b=1$ or $a-b=1$
Case I : $a+b=1\Rightarrow$ Either $a=0$ or $b=0$ then $a^2-b^2=1$ or $-1$
which is not a prime number.
$\therefore$ This case is not possible
Case II : $a-b=1\to a$ and $b$ can be taken anything with $b<a$.
ln this case $a^2-b^2=a+b$ is a prime number.
$\therefore a^2-b^2=a+b.$