Difference between revisions of "1995 AIME Problems/Problem 6"
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== Solution == | == Solution == | ||
− | We know that <math>n^2 = 2^{62}3^{38}</math> must have <math>(62+1)\times (38+1)</math> [[factor]]s by its [[prime factorization]]. If we group all of these factors (excluding <math>n</math>) into pairs that multiply to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} = 1228</math> factors of <math>n^2</math> that are less than <math>n</math>. There are <math>32\times20-1 = 639</math> factors of <math>n</math>, which clearly are less than <math>n</math>, but are still factors of <math>n^2</math>. Therefore, there are <math>1228-639=\boxed{589}</math> factors of <math>n</math> that do not divide <math>n</math>. | + | We know that <math>n^2 = 2^{62}3^{38}</math> must have <math>(62+1)\times (38+1)</math> [[factor]]s by its [[prime factorization]]. If we group all of these factors (excluding <math>n</math>) into pairs that multiply to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} = 1228</math> factors of <math>n^2</math> that are less than <math>n</math>. There are <math>32\times20-1 = 639</math> factors of <math>n</math>, which clearly are less than <math>n</math>, but are still factors of <math>n^2</math>. Therefore, there are <math>1228-639=\boxed{589}</math> factors of <math>n</math> that do not divide <math>n^2</math>. |
== See also == | == See also == |
Revision as of 18:30, 8 November 2009
Problem
Let How many positive integer divisors of are less than but do not divide ?
Solution
We know that must have factors by its prime factorization. If we group all of these factors (excluding ) into pairs that multiply to , then one factor per pair is less than , and so there are factors of that are less than . There are factors of , which clearly are less than , but are still factors of . Therefore, there are factors of that do not divide .
See also
1995 AIME (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 | ||
All AIME Problems and Solutions |