I'm trying to help my husband with his math quiz, but it's been a long time. Any help will be welcome.

Question

{0,1,2,3,4} is the replacement set for n. Find the value of n if:
a. 2 x 4 = n (mod 5)
b 4 + 3 = n (mod 5)
c. 1 - 3 = n (mod 5)

I would appreciate a good explanation more so than just the answer. Thanks.

Answer 1

In mod n arithmetic all operations are carried out modulo n (that is where the mod comes from). Start with counting forward modulo n. You would go 0, 1, 2, 3, 4, 0, 1 , 2, 3, 4, .... etc. Everytime you reach a multiple of 5 you get 0 and continue up.

a) So in normal decimal arithmetic 2x4 = 8, but 8 is outside of the replacement set. So 8 mod 5 is the residue left after dividing 8 by 5, which in this case is 3. Therefore 2 x 4 = 3 mod 5

b) Applying the same logic 4 + 3 = 7 = 2 mod 5

c) And 1 - 3 = -2. To find the mod 5 result you have to imagine now that you are counting backwards. Counting in mod 5 arithmetic backwards from 0 you get 0, 4, 3, 2, 1, 0, 4, 3, 2, 1 etc.

So 1 - 3 = -2 = 3 mod 5

Answer 2

In modular arithmetic, the numbers never exceed the limit - so mod 5, you never have a number more than 4.

2 x 4 = 8 which goes to 3 (mod 5)

4 + 3 = 2

1 - 3 = 3

(with the last - add 3 to each side - 3 + 3 = 1)

Answer 3

Two numbers are congruent if their remainder by division the mod. is the same.Example 27=132(mod5)
a) 2*4= 8 so the remainder dividing by 5 is 3 son n=3
b)4+3=7 so n=2
c) as far as I remember congruence aapplies only for positve integers
If it were 3+1 = 4 n =4