ACT Math: Modular Arithmetic—Working with Remainders and Congruence

Modular arithmetic is arithmetic with remainders. Notation: a≡b (mod n) means "a is congruent to b modulo n," or "a and b have the same remainder when divided by n." Example: 17≡2 (mod 5) because both 17÷5 and 2÷5 give remainder 2. Application: If today is Tuesday (day 2), what day is 17 days later? (2+17)≡19≡4 (mod 7)≡Thursday (day 4). Modular arithmetic shows up in: (1) Cyclic problems (days of week, positions in a circle), (2) Divisibility rules, (3) Pattern repeating. On the ACT, you might see: "A sequence repeats every 5 terms. If you're at term 23, which term's value does it have?" Answer: 23 mod 5=3, so term 23 has the same value as term 3. Modular arithmetic is powerful for "finding patterns" and "predicting cycles."

Key insight: a mod n always gives a remainder from 0 to n-1. That's why modular arithmetic works for cycles—everything repeats within that range.

Mistake 1: Confusing (a mod n) with (a/n). a mod n is the remainder, not the quotient. 17 mod 5=2 (remainder), not 3 (quotient of 17/5). Mistake 2: Forgetting that remainders range from 0 to n-1. 17 mod 7=3, not 17 or any number larger. Mistake 3: Misapplying modular properties. (a+b) mod n=(a mod n+b mod n) mod n. You can't just ignore the second mod n. Mistake 4: Assuming negative numbers don't work in mod. They do. -3 mod 5=2 (because -3=5(-1)+2). Always verify: Is your remainder between 0 and n-1? Does it match the definition?

Checklist: (1) Identify n (the modulus). (2) Calculate the value. (3) Divide by n and find the remainder. (4) Verify remainder is in range [0, n-1]. (5) Apply the result to solve the problem.

Problem 1: 37 mod 6? 37÷6=6 remainder 1. Answer: 1. Problem 2: If a pattern repeats every 8, and you're at position 29, which position's value does it match? 29 mod 8=5. Answer: position 5. Problem 3: What day is 50 days from Tuesday? 50 mod 7=1 (since 50=7×7+1). Day 2+1=day 3. Answer: Wednesday. Problem 4: Find all x where x≡3 (mod 5) and 0For each, show division, find remainder, and verify it's in the correct range.

Daily drill: Solve one modular problem daily. Alternate between finding mod values and applying them to cycles or patterns. Practice the formula (a mod n).

About 1 modular arithmetic question per ACT Math section involves remainders, cycles, or repeating patterns. These feel exotic to many students but are straightforward once you know the concept. Correctly solving a mod problem signals comfort with abstract thinking, making these high-value questions—right answers are rare, and graders reward deep understanding.

Spend 1-2 days on modular arithmetic. Memorize the definition, practice finding remainders, and apply to cycle/pattern problems. By test day, you'll recognize mod questions and solve them confidently.