I just started doing some AMC questions out of a problem-solving book that was lying around my house. I was wondering if you could advise me on how to approach these problems, and give me a hint on this one. The problem I have been facing lately is that even though the solutions look really obvious once I actually read them, I'm completely lost when it comes to solving them on my own. Anyway, here is the problem.

Suppose that the base-8 representation of a perfect square is ab3c, where a is not equal to 0. What is c?

Thanks everyone!!

  • 3
    $\begingroup$ One bit of advice that I picked up early on with these problems and has stayed with me since is this: 1) write out everything you know, and 2) do NOT focus on a single solution - let your mind wander. You want to think of as many viable ways to tackle a problem as there are and then choose one to go with; of course, you get better at this with practice, but those are some fairly general golden rules that I like to work with. So in your problem, starting from the text of your problem, what do you know immediately? $\endgroup$
    – Pockets
    May 25, 2014 at 22:31
  • $\begingroup$ @Pockets that is actually some nice advice! You make mathematics sound more fun that way, even though I always find it fun and challenging sometimes :) $\endgroup$
    – Mr Pie
    Jan 13, 2018 at 22:39

1 Answer 1


Here is the start of my solution to use as a hint if you want:

Write out the given number in its base $8$ representation, $a \cdot 8^3+b \cdot 8^2 +3 \cdot8^1 +c \cdot 8^0$. Now we know this is a perfect square, and further we can write $a \cdot 8^3+b \cdot 8^2 +3 \cdot8^1 +c \cdot 8^0 = (d \cdot8^1+e \cdot 8^0)^2$ with $d,e \in {1,2,...,7}$ (think about why it must have this form, in particular with $d,e$ nonzero and only nonzero coefficients on $8^1,8^0$). From here expand the square and see if you can determine what c must be. As Samuel said in the comments, this is just one line of thought to consider.


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