2006 AIME I Problems/Problem 7
Problem
Find the number of ordered pairs of positive integers such that
and neither
nor
has a zero digit.
Solution
There are numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities for
are when
or
have a 0 in the tens digit, or if the tens digit of
is 0 or 9. Excluding the hundreds, which were counted above already, there are
numbers in every hundred numbers that have a tens digit of 0 or 9, totaling
such numbers. However, the numbers from 1 to 9 and 991 to 999 do not have 0s, so we must subtract
from that to get
. Therefore, there are
such ordered pairs.
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |