Difference between revisions of "2018 AIME II Problems/Problem 3"
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<cmath>3b+6 = n^2</cmath> | <cmath>3b+6 = n^2</cmath> | ||
<cmath>2b+7 = m^3</cmath> | <cmath>2b+7 = m^3</cmath> | ||
− | We can see <math>n</math> is multiple is 3, so let <math>n=3k</math>, then <math>b= 3k^2-2</math>, substitute <math>b</math> into second condition we have <math>m^3=3( | + | We can see <math>n</math> is multiple is 3, so let <math>n=3k</math>, then <math>b= 3k^2-2</math>, substitute <math>b</math> into second condition we have <math>m^3=3(2k^2+1)</math>. Now we know <math>m</math> is multiple of 3, and is odd. Also, <math>m</math> must be smaller than 15 for <math>b</math> to be smaller than 1000. So the only two possible values for <math>m</math> is 3 and 9. Test and they both works. The final answer is <math>10 + 361 =</math> <math>\boxed{371}</math>. -Mathdummy |
==See Also== | ==See Also== |
Revision as of 18:51, 11 March 2019
Contents
Problem
Find the sum of all positive integers such that the base- integer is a perfect square and the base- integer is a perfect cube.
Solution 1
The first step is to convert and into base-10 numbers. Then, we can write and . It should also be noted that .
Because there are less perfect cubes than perfect squares for the restriction we are given on , it is best to list out all the perfect cubes. Since the maximum can be is 1000 and • , we can list all the perfect cubes less than 2007.
Now, must be one of . However, will always be odd, so we can eliminate the cubes of the even numbers and change our list of potential cubes to , and .
Because is a perfect square and is clearly divisible by 3, it must be divisible by 9, so is divisible by 3. Thus the cube, which is , must also be divisible by 3. Therefore, the only cubes that could potentially be now are and .
We need to test both of these cubes to make sure is a perfect square.
If we set equal to , . If we plug this value of b into , the expression equals , which is indeed a perfect square.
If we set equal to , . If we plug this value of b into , the expression equals , which is .
We have proven that both and are the only solutions, so .
Solution 2
The conditions are: We can see is multiple is 3, so let , then , substitute into second condition we have . Now we know is multiple of 3, and is odd. Also, must be smaller than 15 for to be smaller than 1000. So the only two possible values for is 3 and 9. Test and they both works. The final answer is . -Mathdummy
See Also
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.