Episode 8 — Aptitude and Reasoning / 8.7 — HCF and LCM
8.7 Quick Revision -- HCF and LCM
Use this sheet for last-minute revision before exams. Everything on one page.
1. Definitions at a Glance
HCF (Highest Common Factor) = Largest number that divides all given numbers exactly.
Also called: GCD (Greatest Common Divisor), GCF (Greatest Common Factor).
LCM (Least Common Multiple) = Smallest number that is exactly divisible by all given numbers.
2. All Formulas
# Formula
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1 HCF = Product of LOWEST powers of COMMON prime factors
2 LCM = Product of HIGHEST powers of ALL prime factors
3 HCF(a,b) x LCM(a,b) = a x b [ONLY for 2 numbers]
4 LCM(a,b) = (a x b) / HCF(a,b)
5 HCF of fractions = HCF(numerators) / LCM(denominators)
6 LCM of fractions = LCM(numerators) / HCF(denominators)
7 If HCF(a,b) = h, then a = hx, b = hy, where HCF(x,y) = 1
8 LCM = h x x x y (using notation from #7)
9 Smallest number divisible by all = LCM
10 Number leaving remainder r for all divisors = LCM + r
11 Deficit pattern (divisor - remainder = k for all) = LCM - k
12 Largest divisor leaving same unknown remainder = HCF of pairwise differences
13 Largest divisor leaving known remainder r = HCF of (numbers - r)
3. Method Comparison
| Method | Best For | Steps |
|---|---|---|
| Prime Factorization | Small/medium numbers, finding both HCF and LCM | Factorize -> lowest powers (HCF) or highest powers (LCM) |
| Long Division / Euclidean | Large numbers, HCF only | Divide larger by smaller, repeat with remainder, stop at 0 |
| Common Division | LCM of 3+ numbers | Write in row, divide by primes, multiply all divisors |
| Listing Multiples | Very small numbers, quick LCM | List multiples, find first common |
| Product Formula | When HCF is known, need LCM | LCM = (a x b) / HCF |
4. Key Properties
Property Rule
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HCF divides both numbers Always true
Both numbers divide LCM Always true
HCF divides LCM Always true
HCF <= min(a, b) Always true
LCM >= max(a, b) Always true
One divides the other (a | b) HCF = a, LCM = b
Co-prime (HCF = 1) LCM = a x b
Both numbers equal (a = b) HCF = LCM = a
Consecutive integers HCF = 1
Consecutive even integers HCF = 2
HCF must divide LCM Validity check
5. Shortcut Table
| Situation | Shortcut |
|---|---|
| One number divides the other | HCF = smaller, LCM = larger. No calculation needed. |
| Numbers are co-prime | LCM = product. No calculation needed. |
| Know HCF, need LCM | LCM = product / HCF. One division. |
| 3+ numbers, need LCM | Common division method (fastest). |
| Large numbers, need HCF | Euclidean algorithm (avoid factorization). |
| Pairwise differences shortcut | For "same unknown remainder" -- only need HCF of ANY two differences. |
| Numbers in ratio a:b with HCF = h | Numbers are ah and bh. LCM = h x a x b (if HCF(a,b) = 1). |
6. When to Use HCF vs LCM
Use HCF (Maximum / Largest / Greatest)
- Largest tile for a floor
- Maximum number of equal groups / packets
- Largest piece to cut ropes equally
- Largest container to measure liquids exactly
- Greatest number dividing with remainder conditions
- Simplifying fractions (divide by HCF)
Use LCM (Minimum / Smallest / Earliest)
- Bells ringing together / traffic lights
- Meeting at starting point (circular track)
- Smallest number divisible by all
- Scheduling (when events coincide again)
- Minimum rope/rod length for exact cutting
- Day-off problems (when all get day off together)
Memory Aid
HCF = "How Can I Find the biggest divisor?" --> MAXIMUM problems
LCM = "Least Common Meet-up" --> MINIMUM / TIMING problems
7. Common Word Problem Types -- Quick Reference
| Problem Type | Method | Formula |
|---|---|---|
| Bells ring together after ___ | LCM of intervals | LCM(a, b, c, ...) |
| Meet at starting point | LCM of round times | LCM(t1, t2, t3, ...) |
| Largest square tile | HCF of dimensions | HCF(length, width) |
| Number of tiles | Area / tile^2 | (L/HCF) x (W/HCF) |
| Maximum equal groups | HCF of quantities | HCF(q1, q2, q3, ...) |
| Smallest number div by all | LCM | LCM(d1, d2, d3, ...) |
| Smallest with remainder r (same) | LCM + r | LCM(...) + r |
| Smallest with deficit k | LCM - k | LCM(...) - k |
| Largest with same unknown remainder | HCF of differences | HCF(diff1, diff2) |
| Largest with known remainder r | HCF of (num - r) | HCF(n1-r, n2-r, ...) |
| Largest n-digit number div by all | Floor(max/LCM) x LCM | |
| Smallest n-digit number div by all | Ceil(min/LCM) x LCM | |
| How many times in T seconds? | T / LCM + 1 (including start) | |
| Find other number (2 numbers) | HCF x LCM / known | (h x L) / a |
| Find pairs given HCF and LCM | Co-prime pairs with product L/h | HCF(x,y) = 1, xy = L/h |
8. Validity Checks (Exam Traps)
Check 1: HCF must divide LCM.
If HCF = 15 and LCM = 100 --> 100/15 is not integer --> IMPOSSIBLE.
Check 2: HCF must divide both numbers.
If HCF = 7 and one number is 24 --> 24/7 is not integer --> IMPOSSIBLE.
Check 3: LCM must be a multiple of each number.
If LCM = 36 and one number is 8 --> 36/8 is not integer --> IMPOSSIBLE.
Check 4: HCF x LCM = Product (only for 2 numbers).
NEVER use this for 3+ numbers.
Check 5: When finding pairs, x and y must be CO-PRIME.
Factor pair (6, 10) has HCF(6,10) = 2, so REJECT.
9. Fraction Formulas -- Quick Memory Aid
HCF of fractions: H / L (HCF of tops / LCM of bottoms) --> gives SMALL result
LCM of fractions: L / H (LCM of tops / HCF of bottoms) --> gives LARGE result
Think: "HCF is small, so it uses H on top and L on bottom (makes it smaller)"
10. Euclidean Algorithm -- One-Line Summary
HCF(a, b) = HCF(b, a mod b). Repeat until remainder = 0. Last non-zero = HCF.
Example: HCF(462, 132):
462 mod 132 = 66 --> HCF(132, 66) --> 132 mod 66 = 0 --> HCF = 66
11. Number Pair Patterns
Given HCF = h, LCM = L:
Numbers are h*x, h*y
Constraints: HCF(x, y) = 1 and x * y = L/h
Given ratio a:b and HCF = h:
Numbers are a*h, b*h
LCM = h * a * b (provided HCF(a,b) = 1, which should be true if ratio is in simplest form)
Given HCF, LCM, and sum S:
x + y = S/h, x * y = L/h
Solve the quadratic: t^2 - (S/h)t + (L/h) = 0
12. Must-Remember Numbers
LCM(1 through 10) = 2520
LCM(1 through 6) = 60
LCM(1 through 12) = 27720
HCF of any number with 0 = the number itself (by convention)
HCF(0, 0) is undefined
LCM(a, 1) = a (for any a)
HCF(a, 1) = 1 (for any a)
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