Is n congruent to 1 mod 5 if n is congruent to 6 mod 5?

[TheMathNoob]

Homework Statement

if n is congruent to 6 mod 5
then n is congruent to 1 mod 5?

Homework Equations

The Attempt at a Solution

[/B]
This is not a problem. It's a doubt that I have

 

[ehild]

Homework Statement

if n is congruent to 6 mod 5
then n is congruent to 1 mod 5?

Homework Equations

The Attempt at a Solution

[/B]
This is not a problem. It's a doubt that I have

Can you write 6 as congruent to x mod 5? What is x?

 

[TheMathNoob]

Can you write 6 as congruent to x mod 5? What is x?

x is a number between 0 and 5

 

[ehild]

What does it mean that a number n is congruent to x mod 5?

 

[TheMathNoob]

What does it mean that a number n is congruent to x mod 5?

5 divides n-x

 

[ehild]

5 divides n-x

Yes, but you said that x must be between 0 and 5. Which number is x if n=6?

 

[TheMathNoob]

5 divides n-x

Yes, but you said that x must be between 0 and 5. Which number is x if n=6?

I got it by algebra 5 divides n-6 so n-6=5k, n=5(k+1)+1, so 5 divides n-1 which implies n is congruent to 1 mod 5. I am having another inquiry with my friend. He claims that 16 is not congruent to 2 mod 7 because he thinks that a^2 congruent to b mod n implies that b has to be a perfect square. Is that correct?

 

[ehild]

I got it by algebra 5 divides n-6 so n-6=5k, n=5(k+1)+1, so 5 divides n-1 which implies n is congruent to 1 mod 5.

Correct.

I am having another inquiry with my friend. He claims that 16 is not congruent to 2 mod 7 because he thinks that a^2 congruent to b mod n implies that b has to be a perfect square. Is that correct?

a congruent to b mod n implies that a=kn+b. It follows that a²=(kn+b)²=k²n²+2knb+b². If you divide that by n, the remainder is b². But that remainder can be greater than n. It is the case with your example. 16 = 4², and 4 is 4 mod 7, so k=0, and b²=16. You have to do the division further to get 16 = 2*7+2. 16 is congruent to 2 mod 7.