Engaging with Student Questions: Solving Number Theory Puzzles
Written on
Chapter 1: Introduction to Student-Inspired Problems
In this piece, I continue my exploration of intriguing questions posed by my students. Today, we will delve into a captivating problem from number theory. The solution provided by my students was not only creative but also more efficient than my own initial approach. If you’re interested in catching up, you can read about the first question in this series and my inspiration behind it here.
The question presented was:
Before proceeding, I encourage you to attempt solving this on your own.
Section 1.1: The Student’s Solution
As mentioned, the solution my students provided was both clever and concise, although I refined it slightly. We start by expressing the integer 7 as a combination of 2’s and 3’s. This allows us to rewrite the original equation as:
Next, we combine like terms and extract the highest factors from both sides.
We then analyze the parity (even or odd nature) of each factor. Given our constraints of x>3 and y>0, we have:
Caution: One might be tempted to equate even factors with even factors and odd with odd. While this assumption holds in this instance, it’s not universally applicable. For example:
This was where my students’ argument faltered; they didn’t provide justification for their next step. I invite you to ponder, as I did with them, what distinguishes this example from our situation that justifies assuming equality among the even and odd factors. Please share your thoughts in the comments!
Section 1.2: Detailed Comparison of Factors
Returning to our problem, we compare the factors:
Let’s examine these comparisons more closely, starting with the even factors:
Next, we turn to the odd factors:
Ultimately, we arrive at our sole solution: (x,y)=(4,2). I was truly impressed by this solution, and I hope you find it equally intriguing. Along with the challenge I presented earlier, I invite you to consider whether extending the problem to include all integers x and y yields additional solutions. Feel free to share your insights in the comments!
Thanks for reading! I would be grateful if you would consider following my publication, Y(Math), to help broaden the reach of my articles.
Chapter 2: Engaging with More Challenges
Explore a fun challenge to solve an elementary school math question from Singapore and see how it tests your problem-solving skills.
Can you tackle these GMAT questions in just five seconds? Test your speed and accuracy in this engaging math challenge!