Difference between revisions of "2024 AMC 10A Problems/Problem 19"
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+ | {{duplicate|[[2024 AMC 10A Problems/Problem 19|2024 AMC 10A #19]] and [[2024 AMC 12A Problems/Problem 12|2024 AMC 12A #12]]}} | ||
==Problem== | ==Problem== | ||
The first three terms of a geometric sequence are the integers <math>a,\,720,</math> and <math>b,</math> where <math>a<720<b.</math> What is the sum of the digits of the least possible value of <math>b?</math> | The first three terms of a geometric sequence are the integers <math>a,\,720,</math> and <math>b,</math> where <math>a<720<b.</math> What is the sum of the digits of the least possible value of <math>b?</math> | ||
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==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|num-b=18|num-a=20}} | {{AMC10 box|year=2024|ab=A|num-b=18|num-a=20}} | ||
+ | {{AMC12 box|year=2024|ab=A|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 17:38, 8 November 2024
- The following problem is from both the 2024 AMC 10A #19 and 2024 AMC 12A #12, so both problems redirect to this page.
Contents
[hide]Problem
The first three terms of a geometric sequence are the integers and where What is the sum of the digits of the least possible value of
Solution 1
For a geometric sequence, we have , and we can test values for . We find that and works, and we can test multiples of in between the two values. Finding that none of the multiples of 5 divide besides itself, we know that the answer is .
(Note: To find the value of without bashing, we can observe that , and that multiplying it by gives us , which is really close to . ~ YTH)
~eevee9406
Solution 2
We have . We want to find factors and where such that is minimized, as will then be the least possible value of . After experimenting, we see this is achieved when and , which means our value of is , so our sum is .
~i_am_suk_at_math_2
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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 |
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.