Common Multiples
A common multiple of two numbers a and b is any number divisible by both. The smallest is the LCM, and every common multiple is a multiple of that LCM — so the common multiples of a and b are exactly LCM, 2·LCM, 3·LCM, and so on. This single fact answers most CAT questions: to count common multiples of a and b up to N, you do not list anything; you compute floor(N / LCM(a, b)). For example, multiples of both 6 and 8 up to 200 means multiples of LCM(6,8)=24, giving floor(200/24)=8. The kth common multiple is just k × LCM. A frequent trap is treating "common multiple" as a × b; that only works when a and b are coprime (HCF = 1). When they share factors, a × b overcounts, so always reduce to the LCM first. Tie this to HCF using LCM × HCF = a × b to recover one from the other quickly.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Counting in a range
| Multiples of k in [1, N] | floor(N / k) |
|---|---|
| Multiples of k in [A, B] | floor(B/k) − floor((A−1)/k) |
| Common multiples of a, b in [1, N] | floor(N / LCM(a, b)) |
| kth common multiple of a, b | k × LCM(a, b) |
| nth multiple of k | n × k |
Inclusion–exclusion (divisibility)
| Divisible by a OR b | floor(N/a) + floor(N/b) − floor(N/LCM(a,b)) |
|---|---|
| Divisible by a AND b | floor(N / LCM(a, b)) |
| Divisible by NEITHER a nor b | N − [floor(N/a) + floor(N/b) − floor(N/LCM)] |
| Three sets a, b, c (OR) | Σfloor(N/x) − Σfloor(N/LCM pairs) + floor(N/LCM(a,b,c)) |
| Exactly one of a, b | [floor(N/a) − floor(N/LCM)] + [floor(N/b) − floor(N/LCM)] |