CAT Quant · Study & Practice

Multiples

AreaNumber System DifficultyEasy–Moderate CAT weightage1–2 questions directly + a recurring tool inside LCM/HCF, remainders and counting sets

Counting multiples looks deceptively simple, yet it is one of the most reused micro-skills in CAT Number System and Modern Maths. The entire idea rests on a single fact: the count of multiples of k in the range 1 to N is exactly floor(N/k) — the integer part of N divided by k. From that one expression flows a surprising amount of CAT machinery: how many numbers up to 1000 are divisible by 7, how many are divisible by 2 OR 5, how many leave a given remainder, and how many escape a whole set of divisors. The genuinely tricky part is never the floor itself; it is handling "a or b" without double counting, which is where inclusion–exclusion and the LCM step in. This chapter builds the skill cleanly: first the count of common multiples and how LCM controls them, then the full counting toolkit — multiples of one number, of a or b, of a and b, and the "neither" case via complementary counting. Master these and a whole family of set-counting, range and remainder questions collapses into a few quick floor divisions.

Topics

1 Common Multiples 4 worked · 5 practice · 8-Q test Start → 2 Multiple Counting 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.

  • Count of multiples of k up to N is just floor(N/k) — never list, always divide.
  • For a range [A, B]: floor(B/k) − floor((A−1)/k). Use A−1, not A, or you drop the boundary.
  • Common multiples of a and b are multiples of LCM(a,b); count = floor(N/LCM). Do NOT use a×b unless HCF = 1.
  • "a OR b" = floor(N/a) + floor(N/b) − floor(N/LCM(a,b)). Subtracting the LCM term kills the double count.
  • "Neither a nor b" = N − (a OR b count). Complementary counting is faster than direct.
  • "a but not b" = floor(N/a) − floor(N/LCM(a,b)); "exactly one" = (a OR b) − 2·floor(N/LCM).

⚠️ Common mistakes & traps

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

  • Using a × b for common multiples when a and b share factors — only LCM is correct (a×b overcounts).
  • Forgetting the −floor(N/LCM) term in "a OR b", which double counts numbers divisible by both.
  • In a range [A, B], subtracting floor(A/k) instead of floor((A−1)/k) and losing the A endpoint.
  • Confusing "divisible by a and b" (the LCM count) with "divisible by a or b" (inclusion–exclusion).
  • Rounding N/k instead of taking the floor — 1000/7 is 142, not 143.

📈 CAT exam insight & PYQ analysis

In CAT and XAT, pure "count the multiples" questions are rare as standalone items but the skill is embedded everywhere — LCM/HCF problems, remainder and divisibility sets, and Modern Maths counting. The recurring patterns are inclusion–exclusion over two or three divisors ("divisible by 2 or 3 but not 5"), counting in a shifted range, and common-multiple counting tied to an LCM. Difficulty is Easy–Moderate alone but climbs when wrapped inside a set-theory Venn problem or a remainder framework. Prioritise speed and clean floor arithmetic; the marks come from not double counting.

🎴 Flashcards — instant recall

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

Multiples of k in [1, N]?Tap to reveal
floor(N / k)
Multiples of k in [A, B]?Tap to reveal
floor(B/k) − floor((A−1)/k)
Common multiples of a, b up to N?Tap to reveal
floor(N / LCM(a, b))
kth common multiple of a, b?Tap to reveal
k × LCM(a, b)
Divisible by a OR b (count)?Tap to reveal
floor(N/a) + floor(N/b) − floor(N/LCM)
Divisible by a AND b (count)?Tap to reveal
floor(N / LCM(a, b))
Divisible by NEITHER a nor b?Tap to reveal
N − (a OR b count)
Can common multiples = a × b?Tap to reveal
Only when HCF(a,b) = 1
LCM × HCF = ?Tap to reveal
a × b
Multiples of 7 up to 1000?Tap to reveal
142
"a but not b" count?Tap to reveal
floor(N/a) − floor(N/LCM(a,b))
Multiples of both 6 and 8 up to 200?Tap to reveal
floor(200/24) = 8

📌 Quick revision

The whole chapter rests on floor(N/k) — the count of multiples of k in 1 to N. For a range [A, B], use floor(B/k) − floor((A−1)/k). Common multiples of a and b are multiples of LCM(a,b), so count = floor(N/LCM); never use a × b unless they are coprime. For "a OR b" apply inclusion–exclusion: floor(N/a) + floor(N/b) − floor(N/LCM). "Neither" is N minus that OR-count; "a but not b" is floor(N/a) − floor(N/LCM). Watch the traps: take the floor (not a round), keep the −LCM term, and use A−1 at the lower boundary.

Chapter test

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

You have truly mastered Multiples 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