You are Here: Home >< Maths

# Prove 16x(4x+5)(x+1) is a multiple of 48 where x is greater than or equal to 1. watch

Announcements
1. Can someone prove this?

16x(4x+5)(x+1) is a multiple of 48 where x is greater than or equal to 1.

This should be obvious but its not - ive checked it up to x = 10 and it works.

2. (Original post by bnosnehpets)
Can someone prove this?

16x(4x+5)(x+1) is a multiple of 48 where x is greater than or equal to 1.

This should be obvious but its not - ive checked it up to x = 10 and it works.

x has to be one of

3n
3n+1
3n+2

since all numbers are
3. (Original post by TenOfThem)
x has to be one of

3n
3n+1
3n+2

since all numbers are
All numbers are not one of 3n, 3n+1 or 3n+2 (unless n can be 1/3, 2/3, etc.). In that case surely you could say that for any number of ns. For example x is one of 100n, 100n+1, 100n+2, 100n+3, ... and prove that it is a multiple of 1600?

This doesn't make sense to me, can you explain more?
4. (Original post by bnosnehpets)
All numbers are not one of 3n, 3n+1 or 3n+2 (unless n can be 1/3, 2/3, etc.). In that case surely you could say that for any number of ns. For example x is one of 100n, 100n+1, 100n+2, 100n+3, ... and prove that it is a multiple of 1600?

This doesn't make sense to me, can you explain more?
All integers are of the form 3n, 3n+1, 3n+2

I have assumed that x is an integer - otherwise your question does not actually make sense

Not sure what you mean with the 1600 but as that would not work
5. (Original post by TenOfThem)
I have assumed that x is an integer - otherwise your question does not actually make sense

Not sure what you mean with the 1600 but as that would not work
Sorry, I should have specified. x is an interger. Yes, I now understand what you mean by all intergers are of form 3n, 3n+1, 3n+2. However are all intergers not also of the form 4n, 4n+1,4n+2,4n+3 and 5n,5n+1 etc...

I don't understand how this effects the question though. Could you please explain in a bit more detail how what you said effects the proof?

thanks
6. (Original post by bnosnehpets)
Sorry, I should have specified. x is an interger.

I still don't understand though. could you please explain in a bit more detail how what you said effects the proof?

thanks
Have you tried putting the possibilities into the original expression

All should become clear
7. (Original post by bnosnehpets)
Sorry, I should have specified. x is an interger.

I still don't understand though. could you please explain in a bit more detail how what you said effects the proof?

thanks
If x is an integer and you divide it by 3, there are 3 possible results:

remainder 0 i.e. x is divisible by 3 i.e. x = 3n for some integer n
remainder 1 i.e. x = 3n + 1 for some integer n
remainder 2 i.e. x = 3n + 2 for some integer n
8. Sounds like a proof by induction, given the condition on x...
9. (Original post by alexmufc1995)
Sounds like a proof by induction, given the condition on x...
My method is much simpler
10. (Original post by alexmufc1995)
Sounds like a proof by induction, given the condition on x...
hmm ok, that seems strange considering this is not in that section of my course
11. (Original post by TenOfThem)
My method is much simpler
(Original post by bnosnehpets)
hmm ok, that seems strange considering this is not in that section of my course
If you don't know it, try TenOfThem's method. It's more intuitive anyway.
12. (Original post by TenOfThem)
Have you tried putting the possibilities into the original expression

All should become clear
Yes, I have tried up to x=10 and all work.
But I still don't understand how you can prove this. Please can someone talk me through the steps they would take?

Edit: Are you telling me to sub 3n, 3n+1 and 3n+2 into the equation and see if they come out with multiples of 48?
13. (Original post by bnosnehpets)
Yes, I have tried up to x=10 and all work.
But I still don't understand how you can prove this. Please can someone talk me through the steps they would take?
SO you have not put the possibilities in

The steps I would take

Realise that 3 x 16 is 48
Realise that all integers are of one of these forms
3n
3n+1
3n+2
Put these options into the original statement
Look for a factor of 3 in a bracket
Show the 48 is a factor
14. (Original post by TenOfThem)
SO you have not put the possibilities in

The steps I would take

Realise that 3 x 16 is 48
Realise that all integers are of one of these forms
3n
3n+1
3n+2
Put these options into the original statement
Look for a factor of 3 in a bracket
Show the 48 is a factor
Ok thank you! I now understand and all forms work.
15. (Original post by bnosnehpets)
Ok thank you! I now understand and all forms work.
16. (Original post by TenOfThem)
In a rigorous proof, would I have to specify that n>=1 ?
17. (Original post by bnosnehpets)
In a rigorous proof, would I have to specify that n>=1 ?
hmmmm interesting question

I would say that is implied by the fact that x is greater than or equal to 1
18. (Original post by TenOfThem)
hmmmm interesting question

I would say that is implied by the fact that x is greater than or equal to 1
Ok thanks, sorry for the silly comments earlier then, I was overlooking the fact that n could also = 0 (when x = 3n+1 and when x = 3n+2, but not when x = 3n)

19. (Original post by bnosnehpets)
Ok thanks, sorry for the silly comments earlier then, I was overlooking the fact that n could also = 0 (when x = 3n+1 and when x = 3n+2, but not when x = 3n)

No Problem

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: October 2, 2014
Today on TSR

10 instant mood-lifters

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams