Difference between revisions of "2000 AIME I Problems/Problem 1"

(Solution)
(Solution)
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== 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.
 
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.
<cmath>2^1 = 2 | 5^1 = 5</cmath>
+
For n = 1:<cmath>2^1 = 2 , 5^1 = 5</cmath>
<cmath>2^2 = 4 | 5 ^ 2 =25</cmath>
+
n = 2:<cmath>2^2 = 4 , 5 ^ 2 =25</cmath>
<cmath>2^3 = 8 | 5 ^3 = 125</cmath>
+
n = 3:<cmath>2^3 = 8 , 5 ^3 = 125</cmath>
  
 
and so on, until,  
 
and so on, until,  
  
<math>2^8 = 256</math> | <math>5^8 = 390625</math>
+
n = 8:<math>2^8 = 256</math> | <math>5^8 = 390625</math>
  
 
We see that <math>5^8</math> generates the first zero, so the answer is <math>\boxed{008}</math>.
 
We see that <math>5^8</math> generates the first zero, so the answer is <math>\boxed{008}</math>.

Revision as of 17:29, 5 January 2018

Problem

Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$.

Solution

If a factor of $10^{n}$ has a $2$ and a $5$ in its prime factorization, then that factor will end in a $0$. Therefore, we have left to consider the case when the two factors have the $2$s and the $5$s separated, so we need to find the first power of 2 or 5 that contains a 0. For n = 1:\[2^1 = 2 , 5^1 = 5\] n = 2:\[2^2 = 4 , 5 ^ 2 =25\] n = 3:\[2^3 = 8 , 5 ^3 = 125\]

and so on, until,

n = 8:$2^8 = 256$ | $5^8 = 390625$

We see that $5^8$ generates the first zero, so the answer is $\boxed{008}$.

See also

2000 AIME I (ProblemsAnswer KeyResources)
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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