Difference between revisions of "2008 iTest Problems/Problem 9"
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== Problem == | == Problem == | ||
| − | + | Joshua likes to play with numbers and patterns. Joshua's favorite number is <math>6</math> because it is the units digit of his birth year, <math>1996</math>. | |
| − | + | Part of the reason Joshua likes the number 6 so much is that the powers of <math>6</math> all have the same units digit as they grow from <math>6^1</math>: | |
| − | What is the units digit of <math>2008^{2008}</math>? | + | <cmath>\begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*}</cmath> |
| + | However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are | ||
| + | simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, | ||
| + | "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers | ||
| + | and see what he can discover. "See if you can find the units digit of <math>2008^{2008}</math>," Michael challenges. After a little while, | ||
| + | Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of <math>2008^{2008}</math>)? | ||
== Solution == | == Solution == | ||
The statement is equivalent to <math>2008^{2008}\pmod {10}</math>. We can simplify this to <math>8^{2008} \pmod {10}</math>. We see that <math>8^1 \equiv 8 \pmod {10}</math>, <math>8^2 \equiv 4 \pmod {10}</math>, <math>8^3 \equiv 2 \pmod {10}</math>, <math>8^4 \equiv 6 \pmod {10}</math>, and <math>8^5 \equiv 8 \pmod {10}</math>. We see that this pattern will repeat every <math>4</math> terms. Thus, because <math>8^4 \equiv 6 \pmod {10}</math> and the pattern repeats every <math>4</math> terms, <math>8^{2008}\equiv \boxed{6} \pmod {10}</math>. | The statement is equivalent to <math>2008^{2008}\pmod {10}</math>. We can simplify this to <math>8^{2008} \pmod {10}</math>. We see that <math>8^1 \equiv 8 \pmod {10}</math>, <math>8^2 \equiv 4 \pmod {10}</math>, <math>8^3 \equiv 2 \pmod {10}</math>, <math>8^4 \equiv 6 \pmod {10}</math>, and <math>8^5 \equiv 8 \pmod {10}</math>. We see that this pattern will repeat every <math>4</math> terms. Thus, because <math>8^4 \equiv 6 \pmod {10}</math> and the pattern repeats every <math>4</math> terms, <math>8^{2008}\equiv \boxed{6} \pmod {10}</math>. | ||
| − | == See | + | == See Also == |
| + | {{2008 iTest box|num-b=7|num-a=9}} | ||
| + | |||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
Revision as of 00:02, 22 June 2018
Problem
Joshua likes to play with numbers and patterns. Joshua's favorite number is 
because it is the units digit of his birth year, 
.
Part of the reason Joshua likes the number 6 so much is that the powers of all have the same units digit as they grow from 
:
However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are
simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds,
"You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers
and see what he can discover. "See if you can find the units digit of 
," Michael challenges. After a little while,
Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of )?
Solution
The statement is equivalent to 
. We can simplify this to 
. We see that 
, 
, 
, 
, and 
. We see that this pattern will repeat every 
terms. Thus, because and the pattern repeats every terms, 
.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 7 |
Followed by: Problem 9 | |
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