Question

for all integers a,b, and c, if a is congruent to b (mod n), and b is congruent to c (mod n), then a is congruent to c (mod n). Please help prove

Answer 1

This is actually a very common theorem .

Try using :
a-b=nk
b-c=ni
where i and k are integers

To prove:
a-c=nj
where j is an integer

Answer 2

have you tried anything yet?

Answer 3

not yet

Answer 4

if you really want a hint I can tell you
but it would be end of the proof basically

Answer 5

i'd appreciate that hint

Answer 6

try adding the two supposed equations together

Answer 7

so you would have a-c = n(k + i)

Answer 8

and k+i is an integer since integers are closed under addition

Answer 9

so you do have the a-c=nj
which means a is congruent to c mod n

Answer 10

and that proves the theorem?

Answer 11

yep

Answer 12

so j = (k+i)?

Answer 13

yes j can be represented as a sum of integers since j itself is an integer

Answer 14

that makes sense

Answer 15

I would do it without all the extra letters

given n|(a-b) and n|(b-c)

then n|(a-b)+(b-c)=a-c

Answer 16

that is only because you are incredible and are data like-minded

Answer 17

Thanks Zarkon, that was put in a lot simpler terms. Your way helped as well freckles. thanks again