<p>Yeah, we did a little bit of number theory last year, and we never really proved basic addition. Mostly division proofs, actually, ex. if 2|16 and 4|16, prove 8|16.</p>
<p>I had to look up the relation “|” and some basic facts. I hope this isn’t cheating…</p>
<p>Let a,b,c ∈ Z and m ∈ R and a,m≠0:
- if a|b, then a|(bc)
** if a|b, then (am)|(bm)</p>
<p>Assume 8|16.
By <em>, 8|32 (for c = 2).
By *</em>, 4|16 (for m = 1/2) which is given. ∎</p>
<p>Maybe I should try to prove <em>, and *</em>.
proof of <em>:
a|b iff b/a = k1 for some integer k1 (by definition, really). Multiply both sides by integer c. (bc)/a = k2 for some integer k2 (closure of multiplication among integers). Hence a|(bc). ∎
proof of *</em>:
again, a|b iff b/a = k. Multiply both sides by (m/m). We get (bm)/(am) = k(m/m). (m/m) = 1 (multiplicative inverse). So (bm)/(am) = k for some integer k (multiplicative identity). Hence (am)|(bm). ∎</p>
<p>That’s an interesting way to think of it. I’m not sure what my math teacher would have said since he’s a bit of a stickler and has a tendency to be pretty harsh on the implications of the “assume and prove.” (Generally speaking he refused to let us use it on tests…).</p>
<p>This was a pretty easy question as you found out. It got a bit nastier later on, but I’ve really forgotten most of what we did. </p>
<p>Don’t worry we’ll all learn and cry together this year! lol</p>
<p>Your math teacher would be correct. I shouldn’t have done it backwards.
Alternate answer:
16|8 iff 16/8 is an integer. 16/8 = 2, which is an integer. QED b*tches. Of course, this is not pretentious enough and makes the example trivial.</p>
<p>Anyway, the lesson learned is to not assume the result beforehand. Glad we got that out of the way.</p>
<p>Well. Our proof was something along the lines of finding the GCF, proving it divides 16, and then using it as a product.
Something like that. I’ve forgotten. </p>
<p>I’m sure we’ll find out this year man.</p>
<p>LOL…
This thread tricked me in thinking this is AoPS instead of CC lol.
seems like I gonna have fun being a math major.</p>
<p>Seriously? This stuff for AoPS? C’mon man, AoPS is just bloody insane. Since you mentioned AoPS, I’m guessing you’re joining us as a 295er?</p>
<p>Wow, AoPS is pretty cool. I’ve never heard of it.
Which reminds me, we all have to join the math club. Otherwise, we need a means by which we may all collaborate.</p>
<p>AoPS exists as a tool whenever I want to feel like an idiot… >_<
Around AMC time it’s one of the most humbling experiences when people are talking about how they’re disappointed they got 3 or 4 questions wrong and you realize you couldn’t even make the AIME cutoff. </p>
<p>And there’s a math club? Sweet. Although it seems like it’s just painting a target for SAE and Phi Psi, lol.</p>
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<p>I don’t understand this question. Are you trying to show that 8 | 16 without knowing the prime factorization of 16, ie, only knowing that 2 | 16 and 4 | 16? Because you can’t. For example, 2 | 4 and 4 | 4, but 8 does not divide 4.</p>
<p>The actual thing you’re looking for only works when the divisors you’re working with are mutually prime. So if gcd(a,b)=1, then a | c and b | c => ab | c. But if gcd(a,b) is not 1, that doesn’t work.</p>
<p>By the way, math 295 does not have much number theory in it. You’ll learn a little bit about algebraic structures, and that’s fundamental to number theory. However, that’s as far as you go. What you’ll really be doing is an intro to analysis. Imagine calc I with all the theoretical underpinnings.</p>
<p>Also, math 295 is really calc I and II. There’s no 3d calc in it at all, so it’s not at all comparable to math 285.</p>
<p>I made a Facebook group: [Michigan</a> Math 295 Fall '09 | Facebook](<a href=“Facebook Public Group | Facebook”>Facebook Public Group | Facebook) :)</p>
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<p>I used to and I still use AoPS for math stuff (In HS I was really crazy about the AMC/AIME tests, because like the nerd I am, I thought the tests were really brutal yet fun) But I’m not taking 295. But after 285/286 I may take 289 (or the Putnam prep class) just for fun. </p>
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I’m doing the 1:00-2:00p class with Howard, if that helps. Forgot the section number. (usual for me) :D</p>
<p>Woops, I completely forgot about that rule. Yeah, they did need to be mutually prime, we probably did it as if 2|12 and 3|12, prove 6|12.</p>
<p>My bad, and thanks for the correction tetrahedron.</p>
<p>I guess that’s what I get for trying to make up a question from a course I’ve pretty much forgotten.</p>
<p>Hey tetra, since you’re reading this topic (somewhat?), I’ve got a question. Do we need any supplemental material to Spivak, or is Calculus enough?</p>
<p>Does anyone here know that there is a math club that meets every wednesday!!..With free pizza…Math + pizza = Heaven…Cant wait for classes to start…Go Blue</p>
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<p>Spivak is probably all you’ll need. It’s really an excellent book. </p>
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<p>It’s moved to Weds. ? Awesome! It’s been on Thursday afternoon for the last 2 years and I could never make it.</p>
<p>Dude! What time on Wednesday? (I hope it doesn’t interfere with UROP in the evening)</p>
<p>Sorry, I’m pretty sure it’s still 4-5 on Thursdays. The other person must have made a mistake.</p>
<p>[UM</a> Undergraduate Math Club](<a href=“http://www.math.lsa.umich.edu/mathclub/fall2009/index.html]UM”>UM Undergraduate Math Club)</p>
<p>Unless I’m horribly incompetent, they are still on Thursdays.</p>
<p>So, um, taking this class right now, and it’s awesome, but homework takes a loooong time >_<</p>
<p>I’ve spent >10 hours on the homework every week so far!</p>