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# For how many positive integers n is n^2 - 3n + 2 a prime number?

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Math Expert
Joined: 02 Sep 2009
Posts: 54401
For how many positive integers n is n^2 - 3n + 2 a prime number?  [#permalink]

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19 Mar 2019, 00:04
00:00

Difficulty:

45% (medium)

Question Stats:

56% (01:24) correct 44% (01:42) wrong based on 34 sessions

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For how many positive integers n is n^2 - 3n + 2 a prime number?

(A) none
(B) one
(C) two
(D) more than two, but finitely many
(E) infinitely many

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Joined: 04 Jan 2015
Posts: 2803
Re: For how many positive integers n is n^2 - 3n + 2 a prime number?  [#permalink]

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19 Mar 2019, 01:28
1

Solution

Given:
• The number n is a positive integer
• The expression $$n^2 – 3n + 2$$ is a prime number

To find:
• The number of values possible for n

Approach and Working:
If n = even,
• then $$n^2 – 3n + 2$$ = even – even + even = even

if n = odd,
• then $$n^2 – 3n + 2$$ = odd – odd + even = even

Hence, for any value of n, the expression results in an even prime number.

We also know that 2 is the only even prime number.
• Therefore, $$n^2 – 3n + 2 = 2$$
Or, $$n^2 – 3n = 0$$
Or, $$n = 3$$ (as n cannot be 0)

Hence, the correct answer is option B.

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Re: For how many positive integers n is n^2 - 3n + 2 a prime number?  [#permalink]

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19 Mar 2019, 03:54
Bunuel wrote:
For how many positive integers n is n^2 - 3n + 2 a prime number?

(A) none
(B) one
(C) two
(D) more than two, but finitely many
(E) infinitely many

we can reqrite given expression as
n^2 - 3n + 2
n(n-2)-1(n-2)
n=2,1
prime no 2 only 1
IMO B
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Re: For how many positive integers n is n^2 - 3n + 2 a prime number?   [#permalink] 19 Mar 2019, 03:54
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