Question

Let \[P\left( n \right) = {5^n} - {2^n}\]. \[P\left( n \right)\] is divisible by \[3\lambda \], where \[\lambda \] and \[n\] are both odd positive integers. Then, the least value of \[n\] and \[\lambda \] is
(a) 13
(b) 11
(c) 1
(d) 5

Answer 1

Hint: Here, we will substitute the values starting from the least odd positive integer from the given options, to find the value of \[P\left( n \right)\]. Then, we will equate the factors of \[P\left( n \right)\] to \[3\lambda \] to find the value of \[\lambda \]. If the value used satisfies the given equations, where \[\lambda \] and \[n\] are both odd positive integers, then it is the correct option.

Complete step-by-step answer:
We need to find the least value of \[n\] and \[\lambda \].
We can observe that the options have only one value for the least value of both \[n\] and \[\lambda \].
Therefore, the required least value of \[n\] and \[\lambda \] will be equal.
Now, it is given that \[\lambda \] and \[n\] are both odd positive integers.
The least odd positive integer in the given options is 1.
Substituting \[n = 1\] in the given equation \[P\left( n \right) = {5^n} - {2^n}\], we get
\[ \Rightarrow P\left( 1 \right) = {5^1} - {2^1}\]
We know that any number raised to the power 1 is equal to itself.
Therefore, simplifying the expression, we get
\[ \Rightarrow P\left( 1 \right) = 5 - 2\]
Subtracting 2 from 5, we get
\[ \Rightarrow P\left( 1 \right) = 3\]
Thus, we get the value of \[P\left( 1 \right)\] as 3.
Now, we will find the factors of \[P\left( 1 \right)\].
Since the value of \[P\left( 1 \right)\] is 3, the factors of \[P\left( 1 \right)\] are 1 and 3.
This means that \[P\left( 1 \right)\] is divisible by 1 and 3.
It is given that \[P\left( n \right)\] is divisible by \[3\lambda \].
Thus, we will equate the factors of \[P\left( 1 \right)\] to \[3\lambda \], and simplify the equation to find the value of \[3\lambda \].
Equating 1 to \[3\lambda \], we get
\[ \Rightarrow 3\lambda = 1\]
Dividing both sides by 3, we get
\[ \Rightarrow \lambda = \dfrac{1}{3}\]
However, it is given that \[\lambda \] is an odd positive integer.
Thus, \[\lambda \] cannot be equal to \[\dfrac{1}{3}\].
Equating 3 to \[3\lambda \], we get
\[ \Rightarrow 3\lambda = 3\]
Dividing both sides by 3, we get
\[ \Rightarrow \lambda = \dfrac{3}{3}\]
Thus, we get
\[ \Rightarrow \lambda = 1\]
We can observe that both \[\lambda \] and \[n\] are equal to 1.
Therefore, the least value of \[\lambda \] and \[n\] is 1.
Thus, the correct option is option (c).


Note: We do not need to check the other options because we need to find only the least value. Since the odd positive integer 1 satisfies the given information, the other options cannot be the least value, even if they satisfy the given information.
We used the term “factor” in our solution. A number is a factor of another number, if it perfectly divides the number, that is it leaves no remainder. For example, 2 is a factor of 10 because when 10 is divided by 2, the result is 5 and there is no remainder.