Difference between revisions of "2020 AMC 10A Problems/Problem 22"
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<cmath>\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor</cmath> | <cmath>\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor</cmath> | ||
+ | <b>Solution:</b> | ||
Let <math>a = \left\lfloor \frac{998}n \right\rfloor</math>. | Let <math>a = \left\lfloor \frac{998}n \right\rfloor</math>. | ||
Revision as of 20:22, 1 February 2020
Problem
For how many positive integers isnot divisible by ? (Recall that is the greatest integer less than or equal to .)
Solution 1 (Casework)
Expression:
Solution: Let .
Notice that for every integer ,
if is an integer, then the three terms in the expression above must be ,
if is an integer, then the three terms in the expression above must be , and
if is an integer, then the three terms in the expression above must be .
This is due to the fact that , , and share no common factors other than 1. Note that doesn't work; to prove this, all we just have to is substitute for in the expression, to get
This gives us which is divisible by 3.
Now, we test the three cases mentioned above.
Case 1: divides
The first case does not work, as the sum becomes , which is divisible by 3. So, there are only two viable cases:
Case 2: divides
Because divides , the number of possibilities for is the same as the number of factors of , excluding .
=
So, the total number of factors of is .
However, we have to subtract , because the case doesn't work, as mentioned previously.
We now do the same for the third case.
Case 3: divides
=
So, the total number of factors of is .
Again, we have to subtract , for the reason mentioned above in Case 1.
Now that we have counted all of the cases, we add them.
, so the answer is .
~dragonchomper
Video Solution
~IceMatrix
See Also
2020 AMC 10A (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 AMC 10 Problems and Solutions |
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