Multiple Counting

Counting how many numbers in a range satisfy a divisibility condition is pure floor arithmetic. In [1, N], the count of multiples of k is floor(N/k); in a general range [A, B] it is floor(B/k) − floor((A−1)/k). For "divisible by a OR b" you must avoid double counting the numbers divisible by both, which gives the inclusion–exclusion formula floor(N/a) + floor(N/b) − floor(N/LCM(a,b)). For "divisible by NEITHER", subtract that OR-count from N (complementary counting). "Exactly one of a, b" is the OR-count minus twice the both-count, or equivalently the two single-only pieces added. The CAT favourite is mixing these: "how many numbers up to 1000 are divisible by 2 or 3 but not 5". Build it in layers — compute each floor term, plug into inclusion–exclusion, then subtract or filter. Always read whether the range is inclusive and whether 0 is excluded; CAT ranges almost always start at 1.

✅ Solved examples

✏️ Practice — try these, take hints as needed

📝 Topic test — 8 questions

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