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Mod question

  1. Apr 19, 2016 #1
    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
     
  2. jcsd
  3. Apr 19, 2016 #2

    ehild

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    Can you write 6 as congruent to x mod 5? What is x?
     
  4. Apr 20, 2016 #3
    x is a number between 0 and 5
     
  5. Apr 20, 2016 #4

    ehild

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    What does it mean that a number n is congruent to x mod 5?
     
  6. Apr 20, 2016 #5
    5 divides n-x
     
  7. Apr 20, 2016 #5

    ehild

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    Yes, but you said that x must be between 0 and 5. Which number is x if n=6?
     
  8. Apr 20, 2016 #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?
     
  9. Apr 20, 2016 #7

    ehild

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    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.
     
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