Homework Help: Mod question

TheMathNoob · Apr 19, 2016

1. The problem statement, all variables and given/known data
if n is congruent to 6 mod 5
then n is congruent to 1 mod 5?

2. Relevant equations

3. The attempt at a solution

This is not a problem. It's a doubt that I have

ehild · Apr 19, 2016

TheMathNoob said: ↑

1. The problem statement, all variables and given/known data
if n is congruent to 6 mod 5
then n is congruent to 1 mod 5?

2. Relevant equations

3. The attempt at a solution

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 · Apr 20, 2016

ehild said: ↑

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

x is a number between 0 and 5

ehild · Apr 20, 2016

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

TheMathNoob · Apr 20, 2016

ehild said: ↑

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

5 divides n-x

ehild · Apr 20, 2016

TheMathNoob said: ↑

5 divides n-x

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

TheMathNoob · Apr 20, 2016

TheMathNoob said: ↑

5 divides n-x

ehild said: ↑

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 · Apr 20, 2016

TheMathNoob said: ↑

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.

TheMathNoob said: ↑

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.