Very sorry for the disappearing and reappearing blog the past 24 hours or so. Blogger has had some difficulties, but I’ve used it for the better part of a decade without incident, so this one won’t send me running for the hills. I’m assured that the super long post I wrote and posted yesterday about obsessive vocabulary studying will be back shortly. Or I will seriously freak out.
Prize for answering this weekend’s challenge question: I’ll use an image of your choice in a future post. Image can’t be copyrighted, profane, or have anything to do with the Philadelphia Phillies. I reserve the right not to post anything I think sucks.
Your weekend challenge:
If p and q are positive integers, what is p + q?
Put your answers in the comments; I’ll post the solution Monday. Good luck!
UPDATE: Congrats to JD for getting it the long way, and then the short way. Solution below the cut.
It’s funny: I write these questions in a vacuum and sometimes I don’t realize exactly how hard they’ll be until I discuss them with a few people. This one, as these challenge questions go, was especially difficult. I knew it was devious to use 243 in the exponent and the denominator, since the exponent literally could have been anything, but I just couldn’t help myself. So I’m sorry if this one drove you nuts.
There are two insights required to solve this:
- 15243 = 32435243. This is easier to see when you’re dealing with variables (ex: (xy)2 = x2y2), but you can always factor numbers that are raised to exponents, if that’ll help you towards a solution. In this case we’re trying to get to 3p5q; that’s a clue that you’re going to want to factor the 15 out.
- 243 = 35. Yeah. That’s gonna be important.
Now you can cancel the 35 out of the denominator by subtracting 5 from the exponent in 3243 in the numerator. (For a review of this and other exponent rules, click here.)
And there you have it. If p and q are positive integers, then they have to be 238 and 243, respectively. So p + q = 481.
Comments (5)
p + q = 3
Normally I don’t try to solve these until Monday morning but this has been bothering me because my initial try (drastically reducing the 243 and trying to find a pattern) was way off base.
Three notes:
1) 243 = 3^5 ∴ 1/243 = 3^-5
2) 15^243 = (3^243) * (5^243)
3) (3^p)/(3^x) = 3^(p-x)
So, with that in mind, I would divide both side of you equation by 15^243…
1/243 = [(3^p) * (5^q)] / [(3^243) * (5^243)] which reduces to…
1/243 = 3^(p-243) * 5^(q-243)
So, from note #1, we know the right side must work out to 3^-5 in order for the equation to hold. The only way for this to happen is if…
(p-243) = -5 and (q-243) = 0
because then we will have…
1/243 = (3^-5) * (5^0)
so…
p=238 and q=243 ∴
p+q = 481
???
Ok, so that was much more work than necessary since 15^243/243 reduces to…
3^238 * 5^243
which gives you the answer directly instead of going through all the needless steps I went through originally. Funny how the brain works, I did not even see that until a day later…
Yeah, that’s more like it! 🙂 Shoot me an email if you’ve got any great images that should be included in future posts.
481 it is! 15^243 = (3^243)(5^243) And 243 = 3^5, so the problem becomes:
(3^243) (5^243) divided by 3^5. (3^243) divided by (3^5) = 3^238.
So (3^p)(5^q) = (3^238)(5^243). So p + q = 238 + 243, 0r 481.