# 2023 AMC 8 Problems/Problem 22

## 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 (THINKING CREATIVELY!!!)

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## Animated Video Solution

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## Video Solution by harungurcan

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