Question

prove or disprove: the product of any four consecutive integers is divisible by 24.
From what I figure, this is true, but I'm not sure how to prove it. Haven't found a counterexample.

Answer 1

let a be the first integer:
a*(a+1)*(a+2)*(a+3)

Answer 2

what does "is divisible" meant to result in?

Answer 3

or can we do this by induction.. amistre thoughts?

Answer 4

an integer

Answer 5

n/24 is valid for all n if we are just concerned with real values

Answer 6

to disprove you anly need one example thats false

Answer 7

as in n/24 is an integer. and i haven't found a coutnerexample yet to disprove it

Answer 8

-1-2 1 2 = 4

4/24 is not an integer

Answer 9

Oh, sorry only the positive integers...

Answer 10

those aint consecutive lol

Answer 11

4 consecutive integers i thought starting from 1

Answer 12

Induction is for kidds :P

Answer 13

any number less than 24 is going to go off

1*2*3*4/6*4 = 1
2*3*4*5/4*3*2 = 5
3*4*5*6/6*4 = 15

induction, yep

Answer 14

ffm it may be for kids, but it helps

Answer 15

For divisibility by 24 you need to only show the divisibility by 3 and 4 (Excercise: why?)

Now, any 4 consecutive integer must have at-least 2 even numbers and at-least one multiple of 3 hence ... QED

Answer 16

(2n-1)(2n)(2n+1)(4n) 8n^2(4n^2 - 1) 32n^4 - 8n^2
------------------ = -------------- = ------------
24 24 24

Answer 17

4n^4 - n^2
---------- looks less hairy
3

Answer 18

For an inductive proof see here http://www.algebra.com/algebra/homework/word/numbers/Numbers_Word_Problems.faq.question.478538.html

Answer 19

factor out an n^2 to get:

\[n^2(\frac{4n^2-1}{3})\]
since n^2 is always an integer we just got to focus on the other :)

Answer 20

since 4n^2 - 1 = (2n-1)(2n+1)

all we have to do is find a way to prove that the product of 2 consecutive odd intergers is divisible by 3

n=1 ; 1*3 = 3
n=2 ; 3*5 = 15
n=3 ; 5*7 = 35 , but since n^2 = 9 is /3 then we are safe
n=4 ; 7*9 = 63
n=5 ; 9*11 = 99
n=6 ; 11*13 = 143 ; but again, n^2 = 36 so it turns out fine again.


whenever n is a multiple of 3 we get an automatic ok since n^2 is divisible by 3

but i got no idea how to write up a proff for it :/