Difference between revisions of "2009 UNCO Math Contest II Problems/Problem 2"
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== Solution == | == Solution == | ||
| − | { | + | (a) Looking at the units digits, we need the units digit of <math>Q_n</math> to be either <math>0</math> or <math>5</math>. We know that <math>1^n</math> will always have a units digit of <math>1</math>. Looking at <math>2^n</math>, however, cycles every four powers with units digits <math>2, 4, 8,</math> and <math>6</math> in that order. We see that we can only get a units digit of <math>5</math> if we have <math>4</math> as a units digit for <math>2^n</math>, and there is no way to get <math>0</math> as a units digit. Therefore, our answer is <math>\boxed{25}</math> because the four units digits cycle <math>25</math> times in the integers <math>1</math> to <math>100</math>. |
== See also == | == See also == | ||
Revision as of 13:24, 26 November 2016
Problem
(a) Let 
. For how many 
between 
and 
inclusive is 
a multiple of 
?
(b) For how many between and inclusive is 
a multiple of 5?
Solution
(a) Looking at the units digits, we need the units digit of to be either 
or . We know that 
will always have a units digit of . Looking at 
, however, cycles every four powers with units digits 
and 
in that order. We see that we can only get a units digit of if we have 
as a units digit for , and there is no way to get as a units digit. Therefore, our answer is 
because the four units digits cycle 
times in the integers to .
See also
| 2009 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
