CAT Quant · Study & Practice

Modular Arithmetic

AreaNumber System DifficultyModerate–Hard CAT weightage1–3 questions (remainders, cyclicity of units digit, calendar/clock problems)

Modular arithmetic is the maths of remainders, and it quietly powers a big slice of CAT Number System. The idea is simple: instead of caring about the whole number, you only care about what is left over after dividing by a fixed number m, called the modulus. We write a ≡ b (mod m) to say a and b leave the same remainder when divided by m. That single idea collapses enormous computations — the units digit of 7^100, the remainder of 2^500 divided by 7, the day of the week 1,000 days from now — into a few clean steps. CAT loves these because they look terrifying but reward a student who knows the rules. This chapter builds the toolkit in three layers: first the language of congruences and why a ≡ b (mod m) behaves like equality; then the operations — how congruences add, multiply, raise to powers, and how to invert and solve linear congruences such as 3x ≡ 1 (mod 7); and finally the high-yield applications, especially day-of-week and clock problems that turn calendar trivia into a 30-second calculation. Master this and remainder questions stop being guesswork.

Topics

1 Congruences 4 worked · 5 practice · 8-Q test Start → 2 Modular Operations 4 worked · 5 practice · 8-Q test Start → 3 Applications 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Always reduce the base to its smallest residue first; prefer a −1 residue (e.g. 6 ≡ −1 mod 7) so the power becomes a sign.
  • Find a small power that is ≡ 1 (the cycle length), divide the exponent by it, and only the remaining power matters.
  • Fermat: for prime p and gcd(a,p)=1, reduce the exponent modulo (p−1) before computing a^k mod p.
  • Euler: for any m with gcd(a,m)=1, reduce the exponent modulo φ(m); for m = 10, φ(10)=4 explains units-digit cycles of length 4.
  • For a units digit, work mod 10 — every base cycles with period 1, 2 or 4; use exponent mod 4 (treat remainder 0 as the 4th term).
  • Calendar/clock questions are just "N mod 7" (weekday) or "N mod 12" (clock hour); count odd days = 1 per ordinary year, 2 per leap year.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Cancelling a common factor across a congruence without dividing the modulus too — you may only cancel a when gcd(a, m) = 1.
  • Assuming a modular inverse always exists; it exists only when gcd(a, m) = 1, so 2 has no inverse mod 4.
  • Reducing the exponent modulo m instead of modulo (p−1) or φ(m) — the base is reduced mod m, the exponent is not.
  • Forgetting leap years in calendar problems: a leap year contributes 2 odd days, not 1, and century years are leap only if divisible by 400.
  • Misreading the remainder 0 in a cyclicity problem as "the first term"; a remainder of 0 means the last term of the cycle.

📈 CAT exam insight & PYQ analysis

CAT and XAT phrase this topic as remainder and units-digit questions rather than naming "modular arithmetic". Recurring patterns: remainder of a large power divided by a small prime (use Fermat or a −1 residue), units digit via the period-4 cycle, remainders of products and sums of large numbers, and the occasional calendar "what day was/will it be" item. SNAP and IIFT lean harder on direct calendar and clock problems. Difficulty is Moderate–Hard because the setup looks intimidating, but the same three or four reductions solve almost every variant, so accuracy here is a quick percentile gain.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

What does a ≡ b (mod m) mean?Tap to reveal
m divides (a − b); a and b leave the same remainder mod m
a^k mod m equals?Tap to reveal
(a mod m)^k mod m — reduce the base first
When does a modular inverse of a (mod m) exist?Tap to reveal
Exactly when gcd(a, m) = 1
Fermat’s little theorem?Tap to reveal
p prime, gcd(a,p)=1 ⇒ a^(p−1) ≡ 1 (mod p)
Euler’s theorem?Tap to reveal
gcd(a,m)=1 ⇒ a^φ(m) ≡ 1 (mod m)
6^odd (mod 7)?Tap to reveal
−1 ≡ 6, since 6 ≡ −1 (mod 7)
Units-digit cycle length for most bases?Tap to reveal
4 (use exponent mod 4)
Inverse of 3 modulo 7?Tap to reveal
5, since 3×5 = 15 ≡ 1
ax ≡ b (mod m) is solvable when?Tap to reveal
gcd(a, m) divides b
Odd days in an ordinary year vs a leap year?Tap to reveal
1 vs 2
Remainder of 2^500 ÷ 7?Tap to reveal
4 (since 2^3 ≡ 1, 500 ≡ 2 mod 3)
Clock hour after N hours from 12-hour position?Tap to reveal
Work mod 12

📌 Quick revision

Modular arithmetic studies remainders: a ≡ b (mod m) means m divides (a − b). Congruences add, multiply and raise to powers like equations, so always reduce the base to its smallest (often −1) residue first. Kill big exponents by finding a power that is ≡ 1, or use Fermat (a^(p−1) ≡ 1 mod p) and Euler (a^φ(m) ≡ 1) to shrink the exponent. A modular inverse exists only when gcd(a, m) = 1; ax ≡ b (mod m) is solvable when gcd(a,m) divides b. Calendar problems reduce to N mod 7 (1 odd day per ordinary year, 2 per leap year); clock problems reduce to N mod 12. Watch the traps: don’t cancel without gcd = 1, and reduce the exponent mod (p−1) or φ(m), not mod m.

Chapter test

📝 Modular Arithmetic — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Modular Arithmetic when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (3 topics)3/3
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards