CAT Quant · Study & Practice

Remainders

AreaNumber System DifficultyHard CAT weightage1–3 questions (a recurring high-difficulty Number System favourite in CAT/XAT)

Remainders sit at the top of every CAT aspirant’s "high-effort, high-reward" list. The questions look intimidating — find the remainder when 17^256 is divided by 13, or the last two digits of 7^2024 — but they all collapse to one idea: instead of computing the giant number, reduce it modulo the divisor at every step. Once you think in remainders rather than full values, a 200-digit power becomes a one-line calculation. CAT and XAT lean on this topic precisely because brute force is impossible and the smart method is invisible to the unprepared. This chapter builds the full toolkit. We start with the remainder theorem for products and powers (the core habit of replacing a number by its remainder), then negative remainders (treating a remainder as −1 or −2 to kill a power instantly), and finish with the three named theorems that turn ugly exponents into 1: Fermat’s little theorem, Euler’s theorem, and Wilson’s theorem for factorials. Every method comes with worked CAT-style examples, the fastest exam route, and the traps — sign errors, non-coprime divisors, and misread cycle lengths — that quietly cost marks.

Topics

1 Remainder Theorem 4 worked · 5 practice · 8-Q test Start → 2 Negative Remainders 4 worked · 5 practice · 8-Q test Start → 3 Wilson & Fermat 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.

  • Never carry the full number — reduce every base and factor to its remainder before multiplying or raising to a power.
  • Nudge the base to the nearest multiple of n: if it is ≡ −1, the answer is decided by the parity of the exponent ((−1)^even = 1, (−1)^odd = −1).
  • Fermat: for prime p, reduce the exponent mod (p−1) since a^(p−1) ≡ 1 when gcd(a,p)=1.
  • Euler for any modulus: a^φ(n) ≡ 1 (mod n) when gcd(a,n)=1; for last two digits use mod 100, φ(100)=40.
  • Wilson for factorials by a prime p: (p−1)! ≡ −1 and (p−2)! ≡ 1 (mod p).
  • Finish by converting any negative remainder to standard form: add n (e.g. −1 mod 7 → 6).

⚠️ Common mistakes & traps

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

  • Applying Fermat or Euler when the base and modulus are NOT coprime — the theorems then fail.
  • Leaving the answer as a negative remainder (−1) instead of converting to the positive form (n − 1).
  • Reducing the exponent mod p instead of mod (p−1) in Fermat (the cycle length is p−1, not p).
  • Mis-computing φ(n): φ(100) = 40, not 50; use n·∏(1 − 1/p) over DISTINCT primes.
  • Forgetting that the divisor must be prime for Wilson — (n−1)! ≡ −1 only holds for prime n.

📈 CAT exam insight & PYQ analysis

Remainders are a perennial Number System theme in CAT and a XAT favourite, usually one or two genuinely hard questions per paper. The recurring patterns are large powers reduced by Fermat/Euler (last-digit and last-two-digit problems), the −1 trick on bases like 9 mod 10 and 16 mod 17, and factorial remainders dressed up with Wilson’s theorem. CAT increasingly combines remainders with cyclicity and the Chinese Remainder idea (splitting a composite modulus into prime-power parts). Prioritise speed on the standard −1 and Fermat reductions, since those free up time for the rarer multi-step combinations that decide the 99-plus percentile.

🎴 Flashcards — instant recall

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

Remainder of a product, in one line?Tap to reveal
Multiply the remainders, then reduce mod n again.
a^k mod n equals?Tap to reveal
(a mod n)^k mod n
If a ≡ −1 (mod n), then a^k ≡ ?Tap to reveal
(−1)^k: 1 if k even, −1 (= n−1) if k odd
Fermat’s little theorem?Tap to reveal
a^(p−1) ≡ 1 (mod p), p prime, gcd(a,p)=1
In Fermat, reduce the exponent mod ?Tap to reveal
p − 1 (the cycle length), not p
Euler’s theorem?Tap to reveal
a^φ(n) ≡ 1 (mod n) when gcd(a,n)=1
φ(100) = ?Tap to reveal
40 (= 100 × ½ × ⅘)
φ(p) and φ(p^k)?Tap to reveal
φ(p) = p−1; φ(p^k) = p^k − p^(k−1)
Wilson’s theorem?Tap to reveal
(p − 1)! ≡ −1 (mod p), p prime
Corollary of Wilson?Tap to reveal
(p − 2)! ≡ 1 (mod p), p prime
Remainder of 16! by 17?Tap to reveal
16 (since 16! ≡ −1 ≡ 16 mod 17)
Key precondition for Fermat/Euler?Tap to reveal
gcd(base, modulus) = 1 (coprime)

📌 Quick revision

Remainders reward reduction over computation: replace every base and factor by its remainder, then multiply or raise to powers (a^k mod n = (a mod n)^k mod n). Push the base to ±1 modulo n so a power is decided by parity. For large exponents use Fermat (a^(p−1) ≡ 1 mod p) or, for any coprime modulus, Euler (a^φ(n) ≡ 1), and reduce the exponent mod (p−1) or φ(n) — last two digits means mod 100 with φ(100) = 40. Factorials by a prime use Wilson: (p−1)! ≡ −1 and (p−2)! ≡ 1. Always check coprimality before applying a theorem, and convert any negative remainder to its positive form by adding n.

Chapter test

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

You have truly mastered Remainders 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