Difference between revisions of "2020 AMC 8 Problems/Problem 17"
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==Solution== | ==Solution== | ||
We list out the factors of <math>2020</math>: <cmath>1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.</cmath> Of these, only <math>1, 2, 4, 5, 101</math> (<math>5</math> of them) do not have more than <math>3</math> factors. Therefore the answer is <math>\tau\left({2020}\right)-5=\boxed{\textbf{(B) }7}</math>. | We list out the factors of <math>2020</math>: <cmath>1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.</cmath> Of these, only <math>1, 2, 4, 5, 101</math> (<math>5</math> of them) do not have more than <math>3</math> factors. Therefore the answer is <math>\tau\left({2020}\right)-5=\boxed{\textbf{(B) }7}</math>. | ||
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| + | ==Solution 2== | ||
| + | The prime factorization of 2020 is <math>2^2\cdot5\cdot101</math>. The total number of factors of <math>2020</math> is given by the product of one more than each of the prime powers which comes out to <math>3\cdot2\cdot2=12</math>. Instead of finding how many factors of <math>2020</math> have more than three factors, we will instead find how many have one, two, or three factors and subtract this number from <math>12</math> to find the answer. The only number which has one factor is <math>1</math>. For a number to have exactly two factors, it must be prime. From the prime factorization of <math>2020</math>, we know that these can only be <math>2,5,</math> and <math>101</math>. For a number to have three factors, it must be a square of a prime. The only square of a prime that is a factor of <math>2020</math> is 4. Thus our list of factors is <math>1,2,4,5,</math> and <math>101</math> which is a total five factors. Thus, the number of factors of <math>2020</math> that have more than three factors is <math>12-5=7 \implies\boxed{\textbf{(B) }7}</math>.<br> | ||
| + | ~ junaidmansuri | ||
| + | |||
==See also== | ==See also== | ||
Revision as of 06:18, 18 November 2020
How many positive integer factors of 
have more than 
factors?
Solution
We list out the factors of : ![\[1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.\]](http://latex.artofproblemsolving.com/f/2/6/f267298afe7942f84ed0d8e8483876d5e4d2278b.png)
Of these, only 
(
of them) do not have more than factors. Therefore the answer is 
.
Solution 2
The prime factorization of 2020 is 
. The total number of factors of is given by the product of one more than each of the prime powers which comes out to 
. Instead of finding how many factors of have more than three factors, we will instead find how many have one, two, or three factors and subtract this number from 
to find the answer. The only number which has one factor is 
. For a number to have exactly two factors, it must be prime. From the prime factorization of , we know that these can only be 
and 
. For a number to have three factors, it must be a square of a prime. The only square of a prime that is a factor of is 4. Thus our list of factors is 
and which is a total five factors. Thus, the number of factors of that have more than three factors is 
.
~ junaidmansuri
See also
| 2020 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
