How this calculator works
Parentheses override precedence. Without them, this calculator treats mod as an operator at the multiplication level, with a left-to-right tie rule. This is a stated notation convention, since mathematical writing does not assign mod a universal parser. Each table row names the effective parentheses before showing the first operation. Residues use a positive modulus and stay non-negative, including for negative dividends. Some programming languages instead return a signed remainder.
Formula
a+b mod m means a+(b mod m). a×b mod m means (a×b) mod m. a mod b mod m means (a mod b) mod m. r=x-m floor(x/m), m>0.
Worked example
10+8 mod 3 evaluates the remainder first: 10+2=12. Adding parentheses around 10+8 gives 18 mod 3=0. For chained remainders, (20 mod 6) mod 4=2, while 20 mod (6 mod 4)=0.
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Frequently asked questions
Does this evaluate arbitrary text?
Choose one of the three templates and enter integers. The tool shows exactly how each supported expression is grouped.
Why can multiplication give different displayed results?
a×(b mod m) has no final reduction. It is congruent to (a×b) mod m, but may be a different integer.
How do negative numbers behave?
With positive modulus m, the residue is always from 0 through m-1. For example, -7 mod 4 is 1.