Proof (1 Viewer)

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does anyone know why the answer is d and not a

 

[ExtremelyBoredUser]

View attachment 37859
does anyone know why the answer is d and not a

counterexample for a)

2 * 4 * 5 = (3(0) + 2)(3(1) + 1)(3(1)+2) = 40 = 3(13) + 1 = 3k + 1

2, 5 do not fit the definition of a,c so not ALL integers

With trying out results like this, try convincing yourself of (d) and find a way to prove it formally

 

[=)(=]

Ahhh I see Tysm

 

[=)(=]

I have a few more proof qs

for this i get to 3q=2p but theres nothing else to do (I put it into prime factors and equated the powers)


I tried using (k-1)k(k+1) using cases of k odd and even but it doesn't come out

 

[ExtremelyBoredUser]

I have a few more proof qs
View attachment 37865
for this i get to 3q=2p but theres nothing else to do (I put it into prime factors and equated the powers)
View attachment 37866
View attachment 37867
I tried using (k-1)k(k+1) using cases of k odd and even but it doesn't come out

Ill skip the "formal proof" thing since thats smthn i assume u can do

1)

Clearly rational, its 3/2. Find a way to prove that its rational. (Hint; "if X is rational then it can be written in the form ... and because there exists ... such this is true then X must be rational")

2)

Contrapositive here. If a divides b then a divides bc. Should be straightforward. I like to thing of divides as "a factor of" since thats less confusing.

3)

product of three consecutive integers can be represented as n(n+1)(n+2). Consider what happens if n were to be divided by 3, it would either divide perfectly ( n=3k) or would have a remainder (n=3k+1,3k+2). Use proof by cases and figure this one out, should be straightforward.