Find the number by which ${\left( {49} \right)^{15}} - 1$ is exactly divisible by :
A. 50
B. 51
C. 29
D. 8

Hint: We will first apply the condition that is if ${x^n} - {a^n}$ is exactly divisible by $x - a$ to find the factor of ${\left( {49} \right)^{15}} - 1$. Also, the factors of $x - a$ will divide the number, ${x^n} - {a^n}$ completely. We will then find the factors of $49 - 1 = 48$ to find the correct option.

Complete step-by-step answer:
We have to find the number from the given options which exactly divides ${\left( {49} \right)^{15}} - 1$
We know that ${x^n} - {a^n}$ is exactly divisible by $x - a$
We can write ${\left( {49} \right)^{15}} - 1$ as ${\left( {49} \right)^{15}} - {1^{15}}$
Then, we can say ${\left( {49} \right)^{15}} - {1^{15}}$ is exactly divisible by $49 - 1 = 48$
Hence, ${\left( {49} \right)^{15}} - 1$ is divisible by 48.
But, we do not have any option as 48.
Also, since 48 divides the given number completely, the factors of 48 will also divide the given number completely.
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 24, 48.
From the given options, we can say 8 divides the given completely.
Hence, option D is correct.

Note: Factors of a number are numbers that divide the given number completely, that is without leaving any remainder. It is not convenient to find the value of ${\left( {49} \right)^{15}} - 1$ and then find its factors, therefore, one must know the property that if ${x^n} - {a^n}$ is exactly divisible by $x - a$ to this question correctly.