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Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5"

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== Solution ==
 
== Solution ==
 
What you have to realize is that the sum of the digits of a number is the number <math>\pmod{9}</math>. We can prove this right now.
 
What you have to realize is that the sum of the digits of a number is the number <math>\pmod{9}</math>. We can prove this right now.
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Any number can be represented in base-10 like this:
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\begin{equation}
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<cmath>N=a_{1}\cdot10^n+a_{2}\cdot10^{n-1}+\cdots+a_{n-1}\cdot10^1+a_n,</cmath>
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\end{equation}
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where <math>0<a_n<10</math>. Now realize <math>10\equiv1 \pmod{9}</math>, so you can use properties of mod to find <math>N \pmod{9}.</math>
 
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<cmath>N \equiv a_1+a_2+\cdots+a_{n-1}+a_n \pmod{9}</cmath>
 
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What is significant here? By repeatedly summing the digits, you are repeatedly looking for the remainder when that sum is divided by 9. This means that the answer will be the original number <math>\pmod{9}</math>. Either by using the Euler Theorem and the fact that <math>\phi(9)=6</math>, or just by finding a pattern, you see that <math>5^6 \equiv 1 \pmod{9}</math>. This means that <math>5^{2014}=(5^6)^{335}\cdot 5^4 \equiv 5^4 \pmod{9}</math>, which, if calculated properly (not that hard to do), gives you a digit of <math>\boxed{4}</math>.
 
  
 
== See also ==
 
== See also ==
 
{{UNM-PNM Math Contest box|year=2014|n=II|num-b=4|num-a=6}}
 
{{UNM-PNM Math Contest box|year=2014|n=II|num-b=4|num-a=6}}
  
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[[Category:Intermediate Number Theory Problems]]
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[[Category:Intermediate Algebra Problems]]

Revision as of 02:05, 17 March 2019

Problem

$5^n$ is written on the blackboard. The sum of its digits is calculated. Then the sum of the digits of the result is calculated and so on until we have a single digit. If $n = 2014$, what is this digit?

Solution

What you have to realize is that the sum of the digits of a number is the number $\pmod{9}$. We can prove this right now. \begin{equation} \end{equation}

See also

2014 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNM-PNM Problems and Solutions
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