MIT 18.02 Multivariable Calculus, Fall 2007 OpenStudy (anonymous):

Hi, i'm trying to learn how to do proper mathematical proofs for my assignments, but i can't seem to grasp the correct logic. Could anyone give me any hints or recommend any good material to help (clear books, software, videos, etc...). @Mathematics OpenStudy (anonymous):

Do you have a specific problem you would like to work out? OpenStudy (anonymous):

yes. It says to - Prove that if a|c and b|c and gcd(a,b) = 1, then ab|c. OpenStudy (anonymous):

Alright. So we are going to do whats called a Direct Proof. This is where we take whats given, and manipulate it to get what we want to prove. First lets actually say what we are given. OpenStudy (anonymous):

We are given:$a\mid c, b\mid c, (a,b)=1$but what does this actually mean? $a\mid c\iff c=ak_1,k_1\in \mathbb Z$this is telling us that a divides c means that c is equal to a times some integer. Likewise:$b\mid c\iff c=bk_2,k_2\in \mathbb Z$ OpenStudy (anonymous):

Im going to use Bezout's Identity for the last part. Bezout's Identity states if the gcd(a,b)=1, then there exist integers x and y such that:$ax+by=1$ So now that ive said what each of the 3 things given to us means, now lets try to think of how i can combine them to prove that:$ab\mid c$ OpenStudy (anonymous):

Taking Bezout's Identity and multiplying by c gives:$ax+by=1\iff acx+bcy=c$ But I know that:$c=ak_1=bk_2$Substituting those values in I obtain:$acx+bcy=c\Longrightarrow abk_2+bak_1=c$ OpenStudy (anonymous):

The left side of the equation is divisible by ab because:$abk_2+bak_1=ab(k_2+k_1)$Therefore, because the left side is divibsle by ab, the right side is as well, and we get:$ab\mid c$ OpenStudy (anonymous):

And that's the final proof? OpenStudy (anonymous):

yes. Every step is either justified, or is just an algebraic step. thats what makes is a solid proof. OpenStudy (anonymous):

Ok. Thanks!

Latest Questions oweifjwoijf: who can help me with history
1 hour ago 0 Replies 0 Medals jade20: Find three quotations from the text where the author presents evidence to support the claim.
1 hour ago 0 Replies 0 Medals oweifjwoijf: who can help me with history
1 hour ago 29 Replies 1 Medal oweifjwoijf: who got zoom and can help me with history?
2 hours ago 0 Replies 0 Medals ASHLEY74G8: An equation is shown below: 4(x u2212 3) u2212 5(x + 1) = 3 Which statement shows a correct next step in solving the equation? (1 point) The equation can be
2 hours ago 0 Replies 0 Medals homersimpson: Honors Assessment Is there a society in a golden age today? For this Honors assessment, your task is to research a civilization (could be a single country)
2 hours ago 0 Replies 0 Medals sammiitate23: Bob Jones bought a new log cabin for \$73,000 at 11% interest for 30 years. Prepare an amortization schedule for the first three periods.
2 hours ago 0 Replies 0 Medals Iddk: At clean cut manufacturing
3 hours ago 1 Reply 0 Medals CynthiaG1: Identify the sentence that contains a verb phrase. A. African violets are an excellent choice for an indoor garden.
3 hours ago 0 Replies 0 Medals HashmiT: trisha
3 hours ago 0 Replies 0 Medals