Difference between revisions of "2022 AMC 10B Problems/Problem 13"
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Through the givens, we can see that <math>p \approx q</math>. | Through the givens, we can see that <math>p \approx q</math>. | ||
− | Thus, <math>31106=2(p^{2}+pq+q^{2})\approx 6p^{2}\\p^2\approx \tfrac{31106}{6}\approx 5200 | + | Thus, <math>31106=2(p^{2}+pq+q^{2})\approx 6p^{2}\\p^2\approx \tfrac{31106}{6}\approx 5200 |
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− | The least prime greater than these two primes is <math>79</math> 7 + 9 <math>\implies \boxed{\textbf{(E) }16}</math> | + | Recall that </math>70^2=4900<math> and </math>80^2=6400<math>. It follows that our primes must be only marginally larger than </math>70<math>, where we conveniently find </math>p=73, q=71<math> |
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+ | The least prime greater than these two primes is </math>79<math> </math>\implies 7 + 9 <math>\implies \boxed{\textbf{(E) }16}</math> | ||
~BrandonZhang202415 | ~BrandonZhang202415 |
Revision as of 23:08, 14 June 2023
Contents
Problem
The positive difference between a pair of primes is equal to , and the positive difference between the cubes of the two primes is . What is the sum of the digits of the least prime that is greater than those two primes?
Solution 1
Let the two primes be and . We would have and . Using difference of cubes, we would have . Since we know is equal to , would become . Simplifying more, we would get .
Now let's introduce another variable. Instead of using and , we can express the primes as and where is and b is . Plugging and in, we would have . When we expand the parenthesis, it would become . Then we combine like terms to get which equals . Then we subtract 4 from both sides to get . Since all three numbers are divisible by 3, we can divide by 3 to get .
Notice how if we had 1 to both sides, the left side would become a perfect square trinomial: which is . Since is too small to be a valid number, the two primes must be odd, therefore is the number in the middle of them. Conveniently enough, so the two numbers are and . The next prime number is , and so the answer is .
~Trex226
Solution 2
Let the two primes be and , with being the smaller prime. We have , and . Using difference of cubes, we obtain . Now, we use the equation to obtain . Hence, Because we have , . Thus, , so . This implies , , and thus the next biggest prime is , so our answer is
~mathboy100
Solution 3 (Estimation)
Let the two primes be and such that and
By the difference of cubes formula,
Plugging in and ,
Through the givens, we can see that .
Thus, $31106=2(p^{2}+pq+q^{2})\approx 6p^{2}\\p^2\approx \tfrac{31106}{6}\approx 5200
Recall that$ (Error compiling LaTeX. Unknown error_msg)70^2=490080^2=640070p=73, q=7179$$ (Error compiling LaTeX. Unknown error_msg)\implies 7 + 9
~BrandonZhang202415 ~SwordOfJustice (small edits)
Video Solution 1
~Education, the Study of Everything
Video Solution by Interstigation
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.