Difference between revisions of "2013 AMC 10A Problems/Problem 19"
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==Problem== | ==Problem== | ||
| − | In base <math>10</math>, the number <math>2013</math> ends in the digit <math>3</math>. In base <math>9</math>, on the other hand, the same number is written as <math>(2676)_{9}</math> and ends in the digit <math>6</math>. For how many | + | In base <math>10</math>, the number <math>2013</math> ends in the digit <math>3</math>. In base <math>9</math>, on the other hand, the same number is written as <math>(2676)_{9}</math> and ends in the digit <math>6</math>. For how many positive integers <math>b</math> does the base-<math>b</math>-representation of <math>2013</math> end in the digit <math>3</math>? |
Revision as of 12:40, 28 June 2013
Problem
In base 
, the number 
ends in the digit 
. In base 
, on the other hand, the same number is written as 
and ends in the digit 
. For how many positive integers 
does the base--representation of end in the digit ?
Solution
We want the integers such that 
is a factor of 
. Since 
, it has 
factors. Since cannot equal 
or , as these cannot have the digit 3 in their base representations, our answer is 
See Also
| 2013 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
