Question

For how many positive integers \[n\], \[{n^3} - 8{n^2} + 20n - 13\] is a prime number?

Answer 1

Hint: Here, we will first replace \[n\] with the first six natural numbers in the given equation and then check if the obtained solutions are a prime number or not. The natural numbers, which give us the prime numbers, are the required answers.

Complete step-by-step answer:
We are given that the equation is \[{n^3} - 8{n^2} + 20n - 13\], where \[n\] is positive integers.

First, we will replace 1 for \[n\] in the given equation then we have

\[
   \Rightarrow {1^3} - 8{\left( 1 \right)^2} + 20 \times 1 - 13 \\
   \Rightarrow 1 - 8 + 20 - 13 \\
   \Rightarrow 21 - 21 \\
   \Rightarrow 0 \\
 \]

Now we have that 0 is not a prime number.

Replacing 2 for \[n\] in the given equation, we get

\[
   \Rightarrow {2^3} - 8{\left( 2 \right)^2} + 20 \times 2 - 13 \\
   \Rightarrow 8 - 32 + 40 - 13 \\
   \Rightarrow 48 - 45 \\
   \Rightarrow 3 \\
 \]

Now we know that 3 is a prime number.

Now, we will replace 3 for \[n\] in the given equation then we have

\[
   \Rightarrow {3^3} - 8{\left( 3 \right)^2} + 20 \times 3 - 13 \\
   \Rightarrow 3 - 72 + 60 - 13 \\
   \Rightarrow 87 - 85 \\
   \Rightarrow 2 \\
 \]

So we have that 2 is a prime number.

Replacing 4 for \[n\] in the given equation then we have

\[
   \Rightarrow {4^3} - 8{\left( 4 \right)^2} + 20 \times 4 - 13 \\
   \Rightarrow 64 - 128 + 80 - 13 \\
   \Rightarrow 144 - 141 \\
   \Rightarrow 3 \\
 \]

Since we know that the number 3 is a prime number.

Now replace 5 for \[n\] in the given equation then we have

\[
   \Rightarrow {5^3} - 8{\left( 5 \right)^2} + 20 \times 5 - 13 \\
   \Rightarrow 125 - 200 + 100 - 13 \\
   \Rightarrow 225 - 213 \\
   \Rightarrow 12 \\
 \]

Since we know that 12 is not a prime number, but an even number.

Replacing 6 for \[n\] in the given equation then we have

\[
   \Rightarrow {6^3} - 8{\left( 6 \right)^2} + 20 \times 6 - 13 \\
   \Rightarrow 216 - 288 + 120 - 13 \\
   \Rightarrow 336 - 301 \\
   \Rightarrow 35 \\
 \]

Since we know that the number 35 is not a prime number, but an odd number.

So, we can say that there are only 3 positive integers, which gives us prime numbers.

Note: While solving this question, students just have to take some positive numbers to find the final answer. Some students only compute the value for 1 or 2 numbers only, which is wrong. So we have to try to solve for at least the first six positive numbers to be sure. After that question is really simple, so we have to be careful while doing calculations.