Question

If $({x^n} - {a^n})$ is completely divisible by $(x - a),$ when
A. $n$ is any natural number
B. $n$ is an even natural number
C. $n$ is any odd natural number
D. $n$ is a prime number

Answer 1

According to given in the question we have to check the given option to check $({x^n} - {a^n})$ is completely divisible by $(x - a)$ So, first of all we have to check the value of n for which the condition is fulfilled. But first of all we have to understand about natural numbers.
Natural number: A natural number is a number or integer which is greater than 0 and starts with the natural number 1 and on incrementing it goes to infinity as ($1,2,3,4,5,6,7,..................)$ and natural numbers are also called the counting numbers.

Formula used: $({a^2} - {b^2}) = (a + b)(a - b)................(1) \\\ ({a^3} - {b^3}) = (a - b)({a^2} + {b^2} + ab)............(2)$

Complete step-by-step answer:
Step 1: Now, first of all we will substitute the value of $n$ is any natural number.
So, as explained in the solution hint we will put any natural number $(n = 1,2,3,4,.......)$ to check whether the condition is true or not.
Step 2: On substituting the value of $(n = 1)$ which is a natural number.
Hence,

$$= \dfrac{{({x^1} - {a^1})}}{{(x - a)}} \\\ = \dfrac{{(x - a)}}{{(x - a)}} \\\ = 1$$

Which is completely divisible by substituting the value of $(n = 1)$ which is a natural number.
Step 3: On substituting the value of $(n = 2)$ which is a natural number.
Hence,
$= \dfrac{{({x^2} - {a^2})}}{{(x - a)}}$
Now, to solve the expression obtained just above we have to use the formula (1) as mentioned in the solution hint.

$$= \dfrac{{(x - a) \times (x + a)}}{{(x - a)}} \\\ = (x + a)$$

Which is completely divisible by substituting the value of $(n = 2)$ which is a natural number.
Step 3: On substituting the value of $(n = 3)$ which is a natural number.
Hence,
$= \dfrac{{({x^3} - {a^3})}}{{(x - a)}}$
Now, to solve the expression obtained just above we have to use the formula (1) as mentioned in the solution hint.

$$= \dfrac{{(x - a)({x^2} + ax + {a^2})}}{{(x - a)}} \\\ = ({x^2} + ax + {a^2})$$

Which is completely divisible by substituting the value of $(n = 3)$ which is a natural number.

Hence, with the help of the formula (1) and (2) we have to obtained the correct option which is (A) $n$ is any natural number

Additional Information: Even natural numbers: The even natural numbers are the numbers divisible by 2 and including 2 and the even natural numbers start with 2 to infinity as $(2,4,6,8,10,...............)$ and they all are positive numbers.
Odd natural numbers: The odd natural numbers are the numbers divisible by 1, and 3 including 1 and the even natural numbers start with 1 to infinity as $(1,3,5,7,9,...............)$ and they all are positive numbers.
Prime numbers: The number which is completely divisible by 1 and itself is known as the prime number and it start with 2 to infinity as $(2,5,7,11,13,17,..............)$

Note: If we substitute the value of $(n = 5)$ which is an odd natural number and a prime number then we can’t divide $({x^n} - {a^n})$ by $(x - a)$ so we can say that $({x^n} - {a^n})$ is not completely divisible by when $n$ is any odd natural number or when $n$ is a prime number
If we substitute the value of $(n = 6)$ which is an even natural number then we can’t divide $({x^n} - {a^n})$ by $(x - a)$ so we can say that $({x^n} - {a^n})$ is not completely divisible by when $n$ is any even natural number.