For how many positive integers n, n³−8n²+20n−13 is a prime number?

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answer is D.

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Detailed Solution

n³−8n²+20n−13, where n is a positive integer.
Substituting n=1,
⇒1³−8(1)²+20×1−13
⇒1−8+20−13
⇒21−21
⇒0
Substituting n=2,
⇒2³−8(2)²+20×2−13
⇒8−32+40−13
⇒48−45
⇒3 (prime number)
Substituting n=3,
⇒3³−8(3)²+20×3−13
⇒3−72+60−13
⇒87−85
⇒2 (prime number)
Substituting n=4,
⇒4³−8(4)²+20×4−13
⇒64−128+80−13
⇒144−141
⇒3 (prime number)
Substituting n=5,
⇒5³−8(5)²+20×5−13
⇒125−200+100−13
⇒225−213
⇒12
Substituting n=6, ⇒6³−8(6)²+20×6−13
⇒216−288+120−13
⇒336−301
⇒35
Hence, we can say that there are only three positive integers that give us prime numbers.

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