Difference between revisions of "2002 AMC 12B Problems/Problem 3"
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+ | {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #3]] and [[2002 AMC 10B Problems|2002 AMC 10B #6]]}} | ||
== Problem == | == Problem == | ||
For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a [[prime]] number? | For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a [[prime]] number? | ||
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\qquad\mathrm{(E)}\ \text{infinitely\ many}</math> | \qquad\mathrm{(E)}\ \text{infinitely\ many}</math> | ||
== Solution == | == Solution == | ||
− | <math>n^2 - 3n + 2 = (n-2)(n-1)</math> | + | Factoring, we get <math>n^2 - 3n + 2 = (n-2)(n-1)</math>. Exactly <math>1</math> of <math>n-2</math> and <math>n-1</math> must be <math>1</math> and the other a prime number. If <math>n-1=1</math>, then <math>n-2=0</math>, and <math>1\times0=0</math>, which is not prime. On the other hand, if <math>n-2=1</math>, then <math>n-1=2</math>, and <math>1\times2=2</math>, which is a prime number. The answer is <math>\boxed{\mathrm{(B)}\ \text{one}}</math>. |
== See also == | == See also == | ||
+ | {{AMC10 box|year=2002|ab=B|num-b=5|num-a=7}} | ||
{{AMC12 box|year=2002|ab=B|num-b=2|num-a=4}} | {{AMC12 box|year=2002|ab=B|num-b=2|num-a=4}} | ||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] |
Revision as of 17:35, 28 July 2011
- The following problem is from both the 2002 AMC 12B #3 and 2002 AMC 10B #6, so both problems redirect to this page.
Problem
For how many positive integers is
a prime number?
Solution
Factoring, we get . Exactly
of
and
must be
and the other a prime number. If
, then
, and
, which is not prime. On the other hand, if
, then
, and
, which is a prime number. The answer is
.
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |