Difference between revisions of "2008 AMC 8 Problems/Problem 22"
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==Solution 2== | ==Solution 2== | ||
− | Instead of finding n, we find <math>x=\frac{n}{3}</math>. We want <math>x</math>, <math>3x</math>, and <math>9x</math> to be three-digit integers. The smallest three-digit integer is <math>100</math>, so that is our minimum value for <math>x</math>, since if <math>x\in\Z</math>, then <math>9x\in\Z</math>. The largest three-digit integer divisible by <math>9</math> is <math>999</math>, so our maximum value is <math>\frac{999}{9}=111</math>. There are <math>12</math> numbers in the closed set <math>[\,100,111]\,</math>, so the answer is <math>\boxed{\textbf{(A)}\ 12}</math>. | + | Instead of finding n, we find <math>x=\frac{n}{3}</math>. We want <math>x</math>, <math>3x</math>, and <math>9x</math> to be three-digit integers. The smallest three-digit integer is <math>100</math>, so that is our minimum value for <math>x</math>, since if <math>x \in \Z</math>, then <math>9x \in \Z</math>. The largest three-digit integer divisible by <math>9</math> is <math>999</math>, so our maximum value is <math>\frac{999}{9}=111</math>. There are <math>12</math> numbers in the closed set <math>[\,100,111]\,</math>, so the answer is <math>\boxed{\textbf{(A)}\ 12}</math>. |
==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}} | {{MAA Notice}} |
Revision as of 20:21, 21 March 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 integers. The smallest three-digit integer is
, so that is our minimum value for
, since if $x \in \Z$ (Error compiling LaTeX. Unknown error_msg), then $9x \in \Z$ (Error compiling LaTeX. Unknown error_msg). The largest three-digit integer divisible by
is
, so our maximum value is
. There are
numbers in the closed set
, so the answer is
.
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.