Computer Science › Math for Programmers

Modulo

The remainder after division; used for wraparound and hashing.

Also known as: modulus, remainder, mod, %

The modulo operation gives the remainder after division. In most languages it’s written %.

7 % 3       # 1   (7 = 2×3 + 1)
10 % 5      # 0   (divides evenly)
5 % 10      # 5   (smaller than the divisor: unchanged)

Uses

Even or odd, and divisibility:

n % 2 == 0          # even
n % 15 == 0         # divisible by 15 (FizzBuzz)

Wrapping around (cycling through a fixed range):

index = (index + 1) % len(items)       # 0, 1, 2, 0, 1, 2 ...
hour = (hour + 5) % 24                 # clock arithmetic
day_of_week = (today + 10) % 7

Picking a bucket, such as for hashing or sharding:

shard = hash(user_id) % 8

Pagination, batching and “every nth item”: if i % 100 == 0: log_progress().

Extracting digits: n % 10 is the last digit, n // 10 removes it.

Gotchas

• Negative numbers differ between languages. In Python, -7 % 3 is 2 (the result takes the divisor’s sign). In JavaScript, C and Java, it’s -1. If you need a non-negative result, check your language, or use ((n % m) + m) % m.
• Division by zero: x % 0 is an error.
• Floats: % works on floating-point numbers, but with the usual rounding surprises (floating point).
• Don’t confuse with percent or integer division (//).
• Bias and clustering: hash % n with a small or poorly chosen n can distribute unevenly.

Python’s divmod(a, b) returns the quotient and remainder together.