# Find the number of n that can be divided by 7

1. Apr 8, 2012

### songoku

1. The problem statement, all variables and given/known data
Given that Un=4n6+n3+5. Find the number of n, where n = {1, 2, 3,... 2009} so that Un can be divided by 7

2. Relevant equations
Don't know

3. The attempt at a solution
Completely blank.....

2. Apr 8, 2012

### mtayab1994

I'd say to use induction.

3. Apr 8, 2012

### micromass

Staff Emeritus
Why don't you check a few n (say for n from 1 to 10) to see what you get?

Another thing you should do is factorize your expression to see if you get something nice.

4. Apr 8, 2012

### Dick

Think about the problem mod 7. Check n=0,1,2,3,4,5,6.

5. Apr 14, 2012

### songoku

I only know mathematical induction is used for proofing. Can it be used to find number of n in this question?

U1=10
U2=269
U3=2948
U4=16453
U5=62630
U6=186845
U7=470944
U8=1049093
U9=2126498
U10=4001005

I can't see any patterns there that can be used to determine which n can be divided by 7....
I also can't factorize the expression. Or maybe you mean using perfect square?

Actually I don't have any knowledge about mod but I'll give it a try.

Let Un=a so for this question a = b (mod 7), where b must be 7k (multiplication of 7)

This is as far as I can go

6. Apr 14, 2012

### Dick

Ok, let's not use the word 'mod'. Can you show $U_{n+7}-U_n$ is divisible by 7? That means $U_n$ is divisible by 7 iff $U_{n+7}$ is divisible by 7.

7. Apr 14, 2012

### songoku

I can show that Un+7 - Unis divisible by 7 by expanding Un+7 and after subtraction, the terms left will all have 7 as factor.

I have several things that I still not clear:
1. How "Un+7 - Unis divisible by 7" be used to find number of n?
2. Why you use Un+7 - Un? Is it because divisible by 7? If the question asking about divisibility by 5 then Un+5 - Un?

Thanks

8. Apr 17, 2012

### Dick

Look. U_0 is not divisible by 7, right? That means U_7, U_14, U_21,... are not divisible by 7. Agree with that?

9. Apr 17, 2012

### songoku

Yes. So if we know which n produces Un that is divisible by 7, then Un+7 will also be divisible by 7 and they will form arithmetic sequence. Do i get your hint correctly?

Then how to find the first n?

Thanks

10. Apr 17, 2012

### Curious3141

Why not simply try testing U1 to U7? I did it, and it took me no more than 5 minutes with a calculator.

EDIT: Or U0 to U6, as Dick suggested, which entails less work. In fact you can even test U-3 to U3, and that would involve the smallest numbers of all, and still give a complete result.

Last edited: Apr 17, 2012
11. Apr 17, 2012

### Dick

There may not be any n. Check each point that might be a start of the arithmetic sequence. Like I said before, try n=0,1,2,3,4,5,6.

12. Apr 17, 2012

### songoku

Ah, I think I get it. Thanks a lot for the help