Prove that r(r^2-1)(3r+2) is divisible by 24

In summary, the conversation discusses a divisibility problem involving the number 24 and how to prove that there are more multiples of 2. Various approaches, such as looking at cases of r being odd or even and using sequences, are suggested to solve the problem. However, the original poster has not responded for ten days.
  • #1
Aryamaan Thakur
11
2
Originally posted in a technical section, so missing the homework template
Can anyone help me with this divisibility problem.

My approach:-
24 = 2*2*2*3
Now,
This can be written as
(r-1)(r)(r+1)(3r+2)
There will be a multiple of 2 and a multiple of 3. But how to prove that there are more multiples of 2.

PLEASE REPLY FAST!
 
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  • #2
How about looking at the cases ##r## odd and ##r## even?
 
  • #3
I would look at four cases of r.
 
  • #4
So for the arbitrary even case, you can let r = 2k, and substitute that in. What do you get there? Then for the arbitrary odd case, let r = (2k-1) and substitute that in. As you have already figured, there will always be a factor of 3 in there, so you only need to look for 8.
 
  • #5
I think a way to solve this might be the write out the sequences for

r
r^2-1
3r-2

and then separate for even and odd so that for example r^2-1 would give
Even - 3, 15, 35, 63
Odd - 0, 8, 24, 48

I played around a little with this method and started to see a pattern in the factors of the numbers at a given index of the sequence
If you can prove that and set of numbers from the same index of all three of the even or odd sequences will always include the prime factors of 24 then that should suffice
 
  • #6
Please, no more hints until the OP comes back. For someone looking for replies REAL FAST, it's odd that he hasn't responded for ten days.
 
  • Like
Likes scottdave
  • #7
Perhaps he is traveling at 299,792,457 m/s
 

1. What does it mean for a number to be divisible by 24?

For a number to be divisible by 24, it means that when divided by 24, the result is a whole number with no remainder. In other words, 24 is a factor of the number.

2. How can we prove that r(r^2-1)(3r+2) is divisible by 24?

We can prove that a number is divisible by 24 by showing that it is divisible by both 3 and 8. In this case, we can factor out a 3 from the expression to get r(3r^2-3)(3r+2). Then, we can factor out an 8 from the remaining terms, leaving us with 24r(r^2-1)(3r+2). Since 24 is a factor of the expression, we can conclude that it is divisible by 24.

3. What is the significance of r(r^2-1)(3r+2) being divisible by 24?

The significance of a number being divisible by 24 is that it can be evenly divided into 24 equal parts. In the context of this expression, it means that 24 is a common factor of all three terms, which can be useful in simplifying or solving equations.

4. Are there any other numbers that r(r^2-1)(3r+2) is also divisible by?

Yes, since 24 is a multiple of 3 and 8, this expression is also divisible by any multiple of 3 and 8. For example, it is also divisible by 48, 72, 96, etc.

5. Can you provide an example to demonstrate the divisibility of r(r^2-1)(3r+2) by 24?

Let's say r = 4. Substituting this value into the expression, we get 4(4^2-1)(3(4)+2) = 4(16-1)(14) = 4(15)(14) = 840. Since 840 is divisible by 24, we can see that r(r^2-1)(3r+2) is indeed divisible by 24.

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