You are Here: Home >< Maths

# Prove that 5 does not divide (a^3) +(a^2) +1 Watch

1. Prove that 5 does not divide (a^3) +(a^2) +1 . This is the question , I'm really struggling with this could anybody check if this is a valid proof I have completed involving mod5 ?

"If (a^3) +(a^2) +1 was divisible by 5 then
(a^3) +(a^2) +1= 0(mod5)
(a^3) +(a^2) = 1(mod5)

The equivalence classes of mod5 are 0,1,2,3 and 4
Subsitiuting these in place of a

(0)+(0) =1(mod5) ,
((1)+(1) =1(mod5)
(8) +(4)=1(mod5)
(27) +(9) = 1(mod5)
(64) +(16)=1(mod5)
[Note all the equals signs above are meant to represent congruency]"

I'm struggling on how to word the next part . If what I've done so far is correct then since none of the numbers on the LHS are equal to 1(mod5) then 5 does not divide the equation .

Hopefully this is almost at a valid proof . Thanks for any help
2. (Original post by Matt5422)
Prove that 5 does not divide (a^3) +(a^2) +1 . This is the question , I'm really struggling with this could anybody check if this is a valid proof I have completed involving mod5 ?

"If (a^3) +(a^2) +1 was divisible by 5 then
(a^3) +(a^2) +1= 0(mod5)
(a^3) +(a^2) = 1(mod5)

The equivalence classes of mod5 are 0,1,2,3 and 4
Subsitiuting these in place of a

(0)+(0) =1(mod5) ,
((1)+(1) =1(mod5)
(8) +(4)=1(mod5)
(27) +(9) = 1(mod5)
(64) +(16)=1(mod5)
[Note all the equals signs above are meant to represent congruency]"

I'm struggling on how to word the next part . If what I've done so far is correct then since none of the numbers on the LHS are equal to 1(mod5) then 5 does not divide the equation .

Hopefully this is almost at a valid proof . Thanks for any help
The step that I've put in bold is wrong. It should be a^3 + a^2 = -1 = 4 (mod 5). Therefore none of the equivalence classes should be congruent to 4 (not 1); and none of them are.
3. (Original post by Matt5422)
Prove that 5 does not divide (a^3) +(a^2) +1 . This is the question , I'm really struggling with this could anybody check if this is a valid proof I have completed involving mod5 ?

"If (a^3) +(a^2) +1 was divisible by 5 then
(a^3) +(a^2) +1= 0(mod5)
(a^3) +(a^2) = 1(mod5)

The equivalence classes of mod5 are 0,1,2,3 and 4
Subsitiuting these in place of a

(0)+(0) =1(mod5) ,
((1)+(1) =1(mod5)
(8) +(4)=1(mod5)
(27) +(9) = 1(mod5)
(64) +(16)=1(mod5)
[Note all the equals signs above are meant to represent congruency]"

I'm struggling on how to word the next part . If what I've done so far is correct then since none of the numbers on the LHS are equal to 1(mod5) then 5 does not divide the equation .

Hopefully this is almost at a valid proof . Thanks for any help
0+0+1=1 (mod 5)
1+1+1=3 (mod 5)
8+4+1=13=3 (mod 5)
27+9+1=37=2 (mod 5)
64+16+1=81=1 (mod 5)
4. (Original post by Raiden10)
0+0+1=1 (mod 5)
1+1+1=3 (mod 5)
8+4+1=13=3 (mod 5)
27+9+1=37=2 (mod 5)
64+16+1=81=1 (mod 5)
I'm not sure exactly how careful you are supposed to be, but that pretty much settles it. You are looking for 0's on the RHS, and since there aren't any, that completes the proof.
5. Thanks for both of your help with correcting my work
6. This is Nottingham University coursework.

So is http://www.thestudentroom.co.uk/show...5#post28699075

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: November 30, 2010
Today on TSR

### Oxford interview invitations

When can you expect yours?

### Official Cambridge interview invite list

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• 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

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.