Difference between revisions of "2008 AMC 8 Problems/Problem 22"
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==Problem== | ==Problem== | ||
− | For how many positive integer values of <math>n</math> are both <math>\frac{n}{3}</math> and <math>3n</math> three-digit whole numbers? | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>For how many positive integer values of <math>n</math> are both <math>\frac{n}{3}</math> and <math>3n</math> three-digit whole numbers?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
<math>\textbf{(A)}\ 12\qquad | <math>\textbf{(A)}\ 12\qquad | ||
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==Solution== | ==Solution== | ||
− | If <math>\frac{n}{3}</math> is a three digit whole number, <math>n</math> must be divisible by 3 and be <math>\ge 100 | + | If <math>\frac{n}{3}</math> is a three-digit whole number, <math>n</math> must be divisible by 3 and be <math>\ge 100\cdot 3=300</math>. If <math>3n</math> is three digits, n must be <math>\le \frac{999}{3}=333</math> So it must be divisible by three and between 300 and 333. There are <math>\boxed{\textbf{(A)}\ 12}</math> such numbers, which you can find by direct counting. |
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+ | ==Solution 2== | ||
+ | Instead of finding n, we find <math>x=\frac{n}{3}</math>. We want <math>x</math> and <math>9x</math> to be three-digit whole numbers. The smallest three-digit whole number is <math>100</math>, so that is our minimum value for <math>x</math>, since if <math>x \in \mathbb{Z^+}</math>, then <math>9x \in \mathbb{Z^+}</math>. The largest three-digit whole number divisible by <math>9</math> is <math>999</math>, so our maximum value for <math>x</math> is <math>\frac{999}{9}=111</math>. There are <math>12</math> whole numbers in the closed set <math>\left[100,111\right]</math> , so the answer is <math>\boxed{\textbf{(A)}\ 12}</math>. | ||
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+ | - ColtsFan10 | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2008|num-b=21|num-a=23}} | {{AMC8 box|year=2008|num-b=21|num-a=23}} | ||
+ | {{MAA Notice}} |
Revision as of 17:27, 28 August 2020
Contents
Problem
For how many positive integer values of are both and three-digit whole numbers?
Solution
If is a three-digit whole number, must be divisible by 3 and be . If is three digits, n must be So it must be divisible by three and between 300 and 333. There are such numbers, which you can find by direct counting.
Solution 2
Instead of finding n, we find . We want and to be three-digit whole numbers. The smallest three-digit whole number is , so that is our minimum value for , since if , then . The largest three-digit whole number divisible by is , so our maximum value for is . There are whole numbers in the closed set , so the answer is .
- ColtsFan10
See Also
2008 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.