2008 AMC 8 Problems/Problem 22
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
, then
. 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 |
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