CAT Quant · Study & Practice

HCF

AreaNumber System DifficultyEasy–Moderate CAT weightage1–2 questions (directly + inside LCM, remainders, ratios and DI word problems)

The HCF (Highest Common Factor, also called GCD) of two or more numbers is the largest number that divides each of them exactly. It is one of the quiet workhorses of CAT Number System — rarely glamorous, but it underpins a whole family of word problems about cutting things into equal pieces, distributing items without remainder, and finding the greatest divisor that leaves fixed remainders. CAT and XAT almost never ask a bare "find the HCF of 24 and 36"; instead they wrap it inside a story: the largest tile that paves a floor exactly, the biggest container that measures out three liquids, the greatest number dividing a set leaving given remainders. This chapter builds HCF from two angles — prime factorization (take the lowest power of each shared prime) and the Euclidean algorithm (repeated division, the fastest method for large numbers). It then turns to applications: HCF of fractions, the remainder trick (HCF of the differences), and the standard tile-and-container problems. Master both the mechanics and the translation from words to "this is an HCF question", and an entire problem type becomes nearly automatic in the exam.

Topics

1 Euclid Method 4 worked · 5 practice · 8-Q test Start → 2 HCF 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.

  • For large numbers skip factoring — run the Euclidean algorithm: HCF(a, b) = HCF(b, a mod b) until the remainder is 0.
  • Pull out an obvious common factor first: HCF(ka, kb) = k × HCF(a, b), so HCF(1230,1450) = 10 × HCF(123,145).
  • Greatest number leaving the SAME remainder = HCF of the pairwise differences, HCF(a−b, b−c, a−c).
  • Greatest number leaving remainders r1, r2, r3 = HCF(a−r1, b−r2, c−r3) — subtract each remainder to make division exact.
  • HCF of fractions = HCF(numerators)/LCM(denominators); the answer is always ≤ the smallest fraction.
  • Largest tile/container = HCF of the dimensions; number of pieces = product of (each dimension ÷ HCF).

⚠️ Common mistakes & traps

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

  • Taking the HIGHEST power of common primes (that gives LCM) instead of the LOWEST power for HCF.
  • Forgetting to subtract the remainders before taking the HCF in "leaves remainder r" problems.
  • Mixing up units in tile/plot problems — convert km, m and cm to the same unit before taking the HCF.
  • For HCF of fractions, taking LCM of numerators or HCF of denominators (it is HCF of numerators over LCM of denominators).
  • Stopping the Euclidean algorithm one step early — the HCF is the LAST non-zero remainder, not the final quotient.

📈 CAT exam insight & PYQ analysis

In CAT and XAT, pure HCF questions are uncommon as standalone items but appear steadily inside the LCM-HCF cluster, remainder problems and word problems about equal distribution or tiling. The recurring patterns are the "greatest number leaving given remainders" trick (HCF of differences), the largest-tile/largest-container application, and HCF-LCM product questions where one is asked given the other and the two numbers. Difficulty is Easy–Moderate alone but rises when fused with co-primes, number-of-factors or ratio framing. Prioritise fast translation from words to "HCF" and fluent use of the Euclidean algorithm on awkward numbers.

🎴 Flashcards — instant recall

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

HCF from prime factorization?Tap to reveal
Product of each common prime to its LOWEST power
Euclidean algorithm rule?Tap to reveal
HCF(a, b) = HCF(b, a mod b) until remainder 0
HCF of fractions formula?Tap to reveal
HCF(numerators) / LCM(denominators)
HCF(a, b) × LCM(a, b) = ?Tap to reveal
a × b
Greatest number dividing a, b, c exactly?Tap to reveal
HCF(a, b, c)
Greatest number leaving the same remainder?Tap to reveal
HCF of the pairwise differences
Greatest number leaving remainders r1, r2, r3?Tap to reveal
HCF(a − r1, b − r2, c − r3)
HCF(ka, kb) = ?Tap to reveal
k × HCF(a, b)
a, b are co-prime means HCF = ?Tap to reveal
1
Largest tile to pave an L×B floor?Tap to reveal
Side = HCF(L, B)
Last non-zero remainder in Euclid gives?Tap to reveal
The HCF
HCF can never exceed?Tap to reveal
The smallest of the given numbers

📌 Quick revision

The HCF is the largest number dividing each given number exactly. Find it by prime factorization (lowest power of each common prime) or, faster for big numbers, the Euclidean algorithm — HCF(a, b) = HCF(b, a mod b), taking the last non-zero remainder. Use HCF(ka, kb) = k × HCF(a, b) to shrink numbers first, and HCF × LCM = a × b for two-number questions. Applications: largest tile/container = HCF of dimensions; greatest number leaving the same remainder = HCF of differences; leaving remainders r1, r2, r3 = HCF(a−r1, b−r2, c−r3); HCF of fractions = HCF(numerators)/LCM(denominators). Watch the traps: lowest (not highest) powers, subtract remainders first, and keep units consistent.

Chapter test

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

You have truly mastered HCF 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 (2 topics)2/2
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards