2000 AIME I Problems/Problem 1: Difference between revisions
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== Problem == | == Problem == | ||
Find the least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least one of these two integers contains the digit <math>0</math>. | |||
== Solution == | == Solution == | ||
If a factor of <math>10^{n}</math> has a <math>2</math> and a <math>5</math> in its [[prime factorization]], then that factor will end in a <math>0</math>. Therefore, we have left to consider the case when the two factors have the <math>2</math>s and the <math>5</math>s separated, so we need to find the first power of 2 or 5 that contains a 0. | |||
For <math>n = 1:</math> <cmath>2^1 = 2 , 5^1 = 5</cmath> | |||
<math>n = 2:</math> <cmath>2^2 = 4 , 5 ^ 2 =25</cmath> | |||
<math>n = 3:</math> <cmath>2^3 = 8 , 5 ^3 = 125</cmath> | |||
and so on, until, | |||
<math>n = 8:</math> <math>2^8 = 256</math> | <math>5^8 = 390625</math> | |||
We see that <math>5^8</math> contains the first zero, so <math>n = \boxed{8}</math>. | |||
== Video Solution== | |||
https://www.youtube.com/watch?v=6cwg9DZ7bX4 | |||
== See also == | == See also == | ||
{{AIME box|year=2000|n=I|before=First Question|num-a=2}} | |||
[[Category:Introductory Number Theory Problems]] | |||
{{MAA Notice}} | |||
Latest revision as of 11:31, 3 January 2025
Problem
Find the least positive integer 
such that no matter how 
is expressed as the product of any two positive integers, at least one of these two integers contains the digit 
.
Solution
If a factor of has a 
and a 
in its prime factorization, then that factor will end in a . Therefore, we have left to consider the case when the two factors have the s and the s separated, so we need to find the first power of 2 or 5 that contains a 0.
For 
![\[2^1 = 2 , 5^1 = 5\]](http://latex.artofproblemsolving.com/8/0/0/800db9e5191bc4e4a5a3b45161e9fb126ee24ca8.png)
![\[2^2 = 4 , 5 ^ 2 =25\]](http://latex.artofproblemsolving.com/3/5/b/35b9714a58da6c3e2555b8c22cf9c2d71234075f.png)
![\[2^3 = 8 , 5 ^3 = 125\]](http://latex.artofproblemsolving.com/b/9/b/b9b6c237c6b9f7c3394ebc7c9a2f05b3d34e07fe.png)
and so on, until,
| 
We see that 
contains the first zero, so 
.
Video Solution
https://www.youtube.com/watch?v=6cwg9DZ7bX4
See also
| 2000 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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