2022 AIME II Problems/Problem 8
Problem
Find the number of positive integers 
whose value can be uniquely determined when the values of 
, 
, and 
are given, where 
denotes the greatest integer less than or equal to the real number 
.
Solution
1. For 
to be uniquely determined, AND 
both need to be a multiple of 
or 
Since either or is odd, we know that either or has to be a multiple of 
We can state the following cases:
1. is a multiple of 
and 
is a multiple of 
2. is a multiple of 
and is a multiple of
3. is a multiple of and is a multiple of
4. is a multiple of and is a multiple of
Solving for each case, we see that there are 
possibilities for cases 1 and 3 each, and 
possibilities for cases 2 and 4 each. However, we overcounted the cases where
1. is a multiple of 
and is a multiple of
2. is a multiple of and is a multiple of
Each case has 
possibilities.
Adding all the cases and correcting for overcounting, we get 
~Lucasfunnyface
Side note: solution does not explain how we found the 20 possibilities, 30, possibilities, etc. It would be great if somebody added that in.
See Also
| 2022 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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