2024 AMC 10B Problems/Problem 24
Contents
[hide]Problem
Let How many of the values , , , and are integers?
Solution (The simplest way)
First, we know that and must be integers since they are both divisible by .
Then Let’s consider the remaining two numbers. Since they are not divisible by , the result of the first term must be a certain number , and the result of the second term must be a certain number . Similarly, the remaining two terms must each be . Their sum is , so and are also integers.
Therefore, the answer is .
Solution 2 (Specific)
Take everything modulo 8 and re-write the entire fraction with denominator 8. This means that we're going to transform the fraction as follows : becomes And in order for to be an integer, it's important to note that must be congruent to 0 modulo 8. Moreover, we know that . We can verify it by taking everything modulo 8 :
If , then -> TRUE If , then -> TRUE If , then it is obvious that the entire expression is divisible by 8. Therefore, it is true. If , then . Therefore, -> TRUE Therefore, there are possible values.
Addendum for certain China test papers : Note that . Therefore, taking everything modulo 8, whilst still maintaining the original expression, gives . This is true.
Therefore, there are possible values. ~elpianista227
Remark
On certain versions of the AMC in China, the problem was restated as follows:
LetHow many of the values , , , and are integers?
By identical reasoning, each term of is an integer, since is even.
Therefore, the answer is .
~iHateGeometry, countmath1
Video Solution 1 by Pi Academy (In Less Than 2 Mins ⚡🚀)
https://youtu.be/Xn1JLzT7mW4?feature=shared
~ Pi Academy
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=24EZaeAThuE
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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.