Episode 8 — Aptitude and Reasoning / 8.7 — HCF and LCM

8.7.b Tips, Tricks, and Shortcuts

1. Quick HCF Using Successive Division

The long division method (Euclidean algorithm) is almost always faster than prime factorization for large numbers. You do not need to know any prime factors -- just keep dividing.

Speed tip: Write only the essential steps.

HCF(391, 493):
  493 = 391 x 1 + 102
  391 = 102 x 3 + 85
  102 = 85 x 1  + 17
  85  = 17 x 5  + 0
  HCF = 17

When to use: Numbers are large (3+ digits), or you cannot easily spot common factors.

2. Quick LCM Using the "Write Multiples" Method

For small numbers (up to ~30), it is often faster to just list multiples mentally.

LCM(6, 8):
  Multiples of 8: 8, 16, 24 <-- 24 is divisible by 6. Done.
  LCM = 24

When to use: One number is small, or both numbers are small. This avoids prime factorization entirely.

3. Quick LCM Using the HCF Shortcut

LCM(a, b) = (a x b) / HCF(a, b)

If you already know the HCF (or can find it quickly), this gives the LCM in one step.

Example:

LCM(18, 24):
  HCF(18, 24) = 6  (by inspection)
  LCM = (18 x 24) / 6 = 432 / 6 = 72

4. Quick LCM: The "Divide and Carry" Speed Trick

For 3 or more numbers, use the common division method, but with this speed trick:

Trick: At each step, only divide numbers that share the current prime. Carry the others down unchanged.

LCM(6, 8, 12, 15):

  2 |  6   8  12  15        (divide those divisible by 2)
  2 |  3   4   6  15
  2 |  3   2   3  15
  3 |  3   1   3  15
  5 |  1   1   1   5
    |  1   1   1   1

  LCM = 2 x 2 x 2 x 3 x 5 = 120

5. HCF of Three or More Numbers -- Chain Method

Instead of prime factorizing all numbers, chain the Euclidean algorithm:

HCF(a, b, c) = HCF(HCF(a, b), c)

Example: HCF(120, 180, 300)

Step 1: HCF(120, 180)
  180 = 120 x 1 + 60
  120 = 60 x 2 + 0
  HCF = 60

Step 2: HCF(60, 300)
  300 = 60 x 5 + 0
  HCF = 60

Answer: HCF(120, 180, 300) = 60

6. Instant Recognition Shortcuts

6.1 When One Number Divides the Other

If a divides b (or b divides a):
  HCF = the smaller number
  LCM = the larger number

Example: HCF(7, 35) = 7, LCM(7, 35) = 35.

6.2 When the Numbers Are Co-prime

If HCF(a, b) = 1:
  LCM = a x b

Example: LCM(8, 15) = 120 (since HCF(8, 15) = 1).

6.3 When the Numbers Are Equal

HCF(a, a) = a
LCM(a, a) = a

6.4 Consecutive Integers

HCF of consecutive integers = 1  (always)
LCM of consecutive integers = their product

Example: HCF(99, 100) = 1, LCM(99, 100) = 9900.

6.5 Consecutive Even Numbers

HCF of two consecutive even numbers = 2

Example: HCF(14, 16) = 2.

7. Word Problem Identification -- What to Use When

This is critical for exam speed. Recognize the pattern instantly.

7.1 Use LCM When:

Clue in ProblemTypeExample
"When will they meet/coincide/ring together again?"Recurring eventsBells, traffic lights
"Smallest number divisible by..."Minimum common multipleNumber theory
"Leaves remainder r when divided by..."LCM + remainderDivision problems
"After how many days will they have a day off together?"SchedulingWork schedules
"Starting point" or "meet at the start"Circular tracksRunning
"Minimum length/quantity to cut into exact pieces"MeasurementRope/rod cutting

7.2 Use HCF When:

Clue in ProblemTypeExample
"Largest number that divides..."Greatest divisorFactoring
"Maximum size of identical pieces"Cutting/tilingFloor tiles, ropes
"Maximum number of equal groups"DistributionBaskets, teams
"Largest measure that measures exactly"MeasurementContainers
"Same remainder" with "largest divisor"Remainder problemsNumber theory
"Reduce to simplest form"Fraction simplificationFractions

7.3 Memory Trick

HCF --> think "Highest" --> used for MAXIMUM problems (largest tile, biggest group)
LCM --> think "Least"   --> used for MINIMUM problems (earliest time, smallest number)

Seems counterintuitive? Remember:
  HCF is the HIGHEST factor, so it gives the MAXIMUM divisor/group.
  LCM is the LEAST multiple, so it gives the MINIMUM common event time.

8. Shortcut for Remainder-Based Problems

8.1 Same Remainder Given

Subtract the remainder from each number, then take HCF.

Example: Largest number dividing 29, 47, 77 leaving remainder 5.

29 - 5 = 24,  47 - 5 = 42,  77 - 5 = 72
HCF(24, 42, 72) = 6

8.2 Same Remainder NOT Given

Take pairwise differences, then HCF of all differences.

Example: Largest number dividing 62, 132, 237 leaving same remainder.

132 - 62  = 70
237 - 132 = 105
237 - 62  = 175

HCF(70, 105, 175) = 35

Speed tip: You only need HCF of any two differences (the third is automatically covered since it is the sum of the other two).

HCF(70, 105) = 35.  Done.

9. Common Exam Traps and How to Avoid Them

Trap 1: Applying HCF x LCM = Product for 3+ Numbers

WRONG: HCF(a,b,c) x LCM(a,b,c) = a x b x c

This formula works ONLY for exactly two numbers.

Trap 2: Forgetting That HCF Must Divide LCM

Question: "HCF of two numbers is 15 and LCM is 100. Is this possible?"
Check: 100 / 15 = 6.67 (not integer)
Answer: NO. Such a pair does not exist.

Trap 3: Forgetting Co-prime Condition When Finding Number Pairs

Given HCF = 6, LCM = 360.
Numbers are 6x and 6y where HCF(x, y) = 1 and x * y = 60.
Pair (6, 10) gives x*y = 60 but HCF(6, 10) = 2 != 1. REJECT.

Trap 4: Not Reading Carefully -- "Exactly Divisible" vs "Leaves Remainder"

"Exactly divisible" --> Answer = LCM
"Leaves remainder r" --> Answer = LCM + r
"Leaves remainder (divisor - k)" --> might be LCM - k pattern

Trap 5: Confusing HCF and LCM in Fraction Formulas

HCF of fractions: HCF of tops / LCM of bottoms  (H/L)
LCM of fractions: LCM of tops / HCF of bottoms  (L/H)

Memory: "HCF is conservative -- uses HCF on top, LCM on bottom (makes smaller)"

10. Speed Calculation Tips

10.1 Prime Factorize by Inspection

Train yourself to factorize instantly:

Quick reference:
  72  = 8 x 9 = 2^3 x 3^2
  120 = 8 x 15 = 2^3 x 3 x 5
  180 = 4 x 45 = 2^2 x 3^2 x 5
  360 = 8 x 45 = 2^3 x 3^2 x 5
  240 = 16 x 15 = 2^4 x 3 x 5
  504 = 8 x 63 = 2^3 x 3^2 x 7

10.2 Use Divisibility Rules

Divisible by 2: last digit even
Divisible by 3: digit sum divisible by 3
Divisible by 4: last two digits divisible by 4
Divisible by 5: ends in 0 or 5
Divisible by 6: divisible by both 2 and 3
Divisible by 8: last three digits divisible by 8
Divisible by 9: digit sum divisible by 9

10.3 When HCF Is Obvious by Inspection

HCF(36, 54) --> both even, both divisible by 9... try 18.
  36/18 = 2, 54/18 = 3.  HCF(2,3) = 1.  So HCF = 18.

The inspection trick: Divide both by any common factor you can see. If the quotients are co-prime, that common factor is the HCF.

11. Summary of Shortcuts

1.  Large numbers --> Euclidean algorithm (not prime factorization)
2.  Small numbers --> List multiples mentally for LCM
3.  Know HCF?    --> LCM = product / HCF
4.  3+ numbers   --> Common division method for LCM
5.  One divides other --> HCF = smaller, LCM = larger
6.  Co-prime     --> LCM = product (HCF = 1)
7.  Bells/schedule --> LCM
8.  Tiles/groups --> HCF
9.  Same remainder given --> Subtract remainder, then HCF
10. Same remainder unknown --> HCF of pairwise differences
11. HCF must divide LCM (validity check)
12. HCF x LCM = product (ONLY for 2 numbers)

Next: 8.7.c Solved Examples