Question:
If $a$ and $ b$ are positive integers such that $a^2 - b^2$ is a prime number, then $a^2 - b^2$ is
If $a$ and $ b$ are positive integers such that $a^2 - b^2$ is a prime number, then $a^2 - b^2$ is
Updated On: May 11, 2024
- $a+b$
- $a - b$
- $ab$
- 1
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The Correct Option is A
Solution and Explanation
$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 : $a - b = 1 \rightarrow a$ and $b$ can be taken anything with $b < a$.
ln this case $a^2 - b^2 = a + b$ is a prime number.
$\therefore \:\:\: a^2 - b^2 = a + b.$
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 : $a - b = 1 \rightarrow a$ and $b$ can be taken anything with $b < a$.
ln this case $a^2 - b^2 = a + b$ is a prime number.
$\therefore \:\:\: a^2 - b^2 = a + b.$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.