Modular reduction is a widely used arithmetic operation. I found many "mod" related words such as
Can anyone explains the difference among these words? Please give examples or idioms.
A modular operation is an operation done modulo some modulus.
If you use the word "mod" by itself, it should be used as an abbreviation of "modulo" rather than as an abbreviation of either of the other two words. Something like "n mod p" as an abbreviation for "n modulo p" is common but "the mod is p" rather than "the modulus is p" is awkward.
I often find it used as e.g.
It can plainly be seen that 6 = 9 (mod 3).
Talking about the remainder itself is not often done.
"The word modulo...is the Latin ablative of modulus " -- Wikipedia