Difference between revisions of "2023 AMC 8 Problems/Problem 22"

Revision as of 14:29, 25 January 2023

Problem

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$ . What is the first term?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10$

Solution 1

Suppose the first $2$ terms were $x$ and $y$ . Then, the next proceeding terms would be $xy$ , $xy^2$ , $x^2y^3$ , and $x^3y^5$ . Since $x^3y^5$ is the $6$ th term, this must be equal to $4000$ . So, $x^3y^5=4000$ . If we prime factorize $4000$ we get $4000 = 5^3 \cdot 2^5$ . We conclude $x=5$ and $y=2$ , which means that the answer is $\boxed{\textbf{(D)}\ 5}$

~MrThinker, numerophile (edits apex304)

Solution 2

In this solution, we will use trial and error to solve. $4000$ can be expressed as $200 \times 20$ . We divide $200$ by $20$ and get $10$ , divide $20$ by $10$ and get $2$ , and divide $10$ by $2$ to get $\boxed{\textbf{(D)}\ 5}$ . No one said that they have to be in ascending order!

Solution by ILoveMath31415926535 and clarification edits by apex304

Video Solution 1 by OmegaLearn (Using Diophantine Equations)

https://youtu.be/SwPcIZxp_gY

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=ms4agKn7lqc

Animated Video Solution

https://youtu.be/tnv1XzSOagA

~Star League (https://starleague.us)

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=2649

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 6: / Line 6: @@
 ==Solution 1==
-Suppose the first <math>2</math> terms were <math>x</math> and <math>y</math>. Then, the next proceeding terms would be <math>xy</math>, <math>xy^2</math>, <math>x^2y^3</math>, and <math>x^3y^5</math>. Since <math>x^3y^5</math> is the <math>6</math>th term, this must be equal to <math>4000</math>. So, <math>x^3y^5=4000 \Rightarrow (xy)^3y^2=4000</math>. If we prime factorize <math>4000</math> we get <math>4000 = 2^5 \times 5^3</math>. We conclude <math>x=5</math>, <math>y=2</math>, which means that the answer is <math>\boxed{\textbf{(D)}\ 5}</math>
+Suppose the first <math>2</math> terms were <math>x</math> and <math>y</math>. Then, the next proceeding terms would be <math>xy</math>, <math>xy^2</math>, <math>x^2y^3</math>, and <math>x^3y^5</math>. Since <math>x^3y^5</math> is the <math>6</math>th term, this must be equal to <math>4000</math>. So, <math>x^3y^5=4000</math>. If we prime factorize <math>4000</math> we get <math>4000 = 5^3 \cdot 2^5</math>. We conclude <math>x=5</math> and <math>y=2</math>, which means that the answer is <math>\boxed{\textbf{(D)}\ 5}</math>
 ~MrThinker, numerophile (edits apex304)

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