Difference between revisions of "1960 AHSME Problems/Problem 33"
Rockmanex3 (talk | contribs) (Solution for Problem 33) |
Rockmanex3 (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||
− | First, note that <math>n</math> does not have a prime number larger than <math>61</math> as one of its factors. Also, note that <math>n</math> does not equal <math>1</math>. | + | First, note that <math>n</math> does not have a [[prime number]] larger than <math>61</math> as one of its factors. Also, note that <math>n</math> does not equal <math>1</math>. |
Therefore, since the prime factorization of <math>n</math> only has primes from <math>2</math> to <math>59</math>, <math>n</math> and <math>P</math> share at least one common factor other than <math>1</math>. Therefore <math>P+n</math> is not prime for any <math>n</math>, so the answer is <math>\boxed{\textbf{(A)}}</math>. | Therefore, since the prime factorization of <math>n</math> only has primes from <math>2</math> to <math>59</math>, <math>n</math> and <math>P</math> share at least one common factor other than <math>1</math>. Therefore <math>P+n</math> is not prime for any <math>n</math>, so the answer is <math>\boxed{\textbf{(A)}}</math>. |
Revision as of 01:23, 13 May 2018
Problem
You are given a sequence of terms; each term has the form where stands for the product of all prime numbers less than or equal to , and takes, successively, the values . Let be the number of primes appearing in this sequence. Then is:
Solution
First, note that does not have a prime number larger than as one of its factors. Also, note that does not equal .
Therefore, since the prime factorization of only has primes from to , and share at least one common factor other than . Therefore is not prime for any , so the answer is .
See Also
1960 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 32 |
Followed by Problem 34 | |
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |