Let a = 67, b = 30, and n = 7

Question

Verify and explain that (a-n)≡(mod b)

Answer 1

Louie,

 

When you see the word "mod" it means you are working as if on a clock.  MOD(30) means you are working on a 30 hour clock.

 

So 67-7 = 60, and on a 30 hour clock every time you go to 30 you arrive at zero.  Just as, on a our common 12 hour clock, every 12 hours both hands stand straight up and point to 12, which is the same as zero.  

 

So on a 30 hour clock 30 is the same as zero, and so is 60.

 

So we write 67 - 7 ≡ 0 mod(30) or alternatively 67 - 7 = 0 mod(30).   You can also say 67 ≡ 7 mod(30).