Difference between revisions of "2024 AMC 10B Problems/Problem 24"
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− | We can rewrite the expression as <cmath>\frac{1}{8}\cdot m \cdot (m^7+m^3+2m+4).</cmath> If <math>m</math> is even, then <math>m</math> gives a factor of <math>2</math> and <math>m^7+m^3+2m+4</math> will give a factor of <math>4,</math> so the result will be an integer. If <math>m</math> is odd, then notice that for any <math>m</math> we have <math>m^2\equiv 1 \pmod{8}.</math> Then <math>m^7+m^3+2m+4\equiv m+m+2m+4 \equiv 4(m+1) \equiv 0 \pmod{8}.</math> So any integer <math>m</math> will result in an integer, meaning the answer is <math>\boxed{\textbf{(E) } | + | We can rewrite the expression as <cmath>\frac{1}{8}\cdot m \cdot (m^7+m^3+2m+4).</cmath> If <math>m</math> is even, then <math>m</math> gives a factor of <math>2</math> and <math>m^7+m^3+2m+4</math> will give a factor of <math>4,</math> so the result will be an integer. If <math>m</math> is odd, then notice that for any <math>m</math> we have <math>m^2\equiv 1 \pmod{8}.</math> Then <math>m^7+m^3+2m+4\equiv m+m+2m+4 \equiv 4(m+1) \equiv 0 \pmod{8}.</math> So any integer <math>m</math> will result in an integer, meaning the answer is <math>\boxed{\textbf{(E) }4}</math>. |
-nevergonnagiveup | -nevergonnagiveup |
Latest revision as of 01:57, 20 December 2024
Contents
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
Solution 3 (Factoring)
We can rewrite the expression as If is even, then gives a factor of and will give a factor of so the result will be an integer. If is odd, then notice that for any we have Then So any integer will result in an integer, meaning the answer is .
-nevergonnagiveup
~flyingkinder123 (minor edits)
Solution 4
Taking a closer look at the terms, we notice that each term builds off of the previous one. There is , , which is equal to , , which is equal to , and , which is equal to . Ultimately, this means that there are only two cases that we need to check for: the case in which is even and the case in which is odd.
If is even, will be an integer, which means the rest of the terms will be an integer. This means in the problem, and will yield an integer result. (For certain Chinese versions, this also proves that is an integer.)
If is odd, will result in . Building off of each term will give you , , and , and summing those up will grant , an integer. This means in the problem, and will also result in integer answers.
In general, all integer will make give an integer answer, but for this question, this will get us to integer values ( for certain Chinese versions of the test).
~unpogged
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 by Pi Academy (Fast and Easy)
https://youtu.be/Xn1JLzT7mW4?si=borSg8xrYLAz7mNY
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.