# Thread: question involving modulus (congrency)

1. ## question involving modulus (congrency)

Hi

I need help with the following question

I have attached my question.

I haven't got a clue on how to start the question.

thanks

2. Originally Posted by cooltowns
Hi

I need help with the following question

I have attached my question.

I haven't got a clue on how to start the question.

thanks

First, you prove that any integer to the 4th power is 0,1,2,or 4 mod 7, and then you reduce the given question modulo 7 and get that there's no solution to it and thus there's no integere solution for it, either.

Tonio

3. can you please show me how to do it, i would appreichate this. I have a few questions which are similar to this. So if i have fully worked out solution to one i can try others on my own.

thanks

4. You solve it by doing it! What is $0^2 (mod 7)$? What is $1^2 (mod 7)$? What is $2^2 (mod 7)$? What is $3^2 (mod 7)$? What is $4^2 (mod 7)$? What is $5^2 (mod 7)$? What is What is $6^2 (mod 7)$?

Now reduce $7x^5+ 3y^4= 2$ "(mod 7)". Since 7= 0 (mod 7) that reduces to $3y^4= 3(y^2)(y^2)= 2$ (mod 7). Is there any integer y for which that is true? Replace $y^2$ with 0, 1, 2, and 4 to see.

5. could you please explain how the $3y^4=3(y^2)(y^2)=2(mod7)$

i don't understand where the 2 came from and why you have chosen mod7 in particular.

thanks

6. Obviously:

$7x^5+3y^4 \equiv 3y^4\ (\text{mod}\ 7)$

Now we know that $y^4\equiv 0, 1,2, \text{ or } 4\ (\text{mod} \ 7)$

So what are the possible values of $3y^4\ (\text{mod}\ 7)$ and are any of these congruent to $2 \ (\text{mod}\ 7)$

CB

7. but why 2(mod7). As i understand the mod 7 comes from the equation but could you please explain where the 2 came from ?

thank you

8. Originally Posted by cooltowns
but why 2(mod7). As i understand the mod 7 comes from the equation but could you please explain where the 2 came from ?

thank you
It's the right hand side of your d**n problem:

$7x^5+3y^4=2$

There is a bit of a hint also in the question that $\text{mod 7}$ is involved.

To prove that an equation has no integer solutions it is sufficient to show that it has no solution modulo some integer.

(As a hint in future read your own problem when you post it)

CB

9. thanks for all the replies.

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