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?(adsbygoogle = window.adsbygoogle || []).push({});

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.

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