Im having a bit of hard time understanding how is that two intergers (a and b) divided by a common divisor (m) have the same remainder imply that the difference of (a and b) will aslo be divisible by m? Essentially what im asking is: a [itex]\equiv[/itex] b (mod m) [itex]\Rightarrow[/itex] m|(a-b) the "|" means divides just incase you aren't fimiliar with that symbol. a [itex]\equiv[/itex] b (mod m) says a/m and b/m will have the same remainder. Since, they have the same remainder (a - b) will also be be divisible by m. example 1) 29 [itex]\equiv[/itex] 15 (mod 7) [itex]\Rightarrow[/itex] 7|(29 -15) Why is the difference of 29 -15 also divisible by 7? Is it because when since 29 and 15 have the same reminder means that we are simply taking out factors of 7 and the common reminder from the 29 and 15? 29 - 15 [7(4) + 1] - [ 7(2) + 1] = 7(2) = 14 which is divisible by 7 example 2) 11 [itex]\equiv[/itex] 4 (mod 7) [itex]\Rightarrow[/itex] 7|(11 - 4) 11 - 4 [7(1) + 4] - [7(0) + 4] = 7 which is divisible by 7 Even if my reasoning is correct, please try to explain in your own way. I can do it mathmetically but that problem im having is understanding it.