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whats shottest way of finding the lcm of three numbers

The shortest, most reliable way to find the LCM of three numbers is to use prime factorization or a streamlined division/ladder method —both avoid long lists of multiples and work fast even for big numbers.

Quick Scoop: What Is LCM?

LCM (Least Common Multiple) of three numbers is the smallest positive number that all three can divide into without leaving a remainder.

You use it a lot when adding fractions, finding repeating cycles, or solving timetable-type problems.

Method 1: Super‑fast Prime Factorization

This is usually the shortest way in terms of steps, especially for larger numbers.

Steps

  1. Prime‑factorize each number.
    Break each into primes:

    • Example: 12, 15, 20
      • 12 = 22×32^2\times 322×3
      • 15 = 3×53\times 53×5
      • 20 = 22×52^2\times 522×5

  1. Pick the highest power of each prime.
    From all three factorizations, look at each prime and take the biggest exponent:

    • Prime 2: highest is 222^222
    • Prime 3: highest is 313^131
    • Prime 5: highest is 515^151
  1. Multiply those together.
    LCM=22×3×5=4×3×5=60LCM=2^2\times 3\times 5=4\times 3\times 5=60LCM=22×3×5=4×3×5=60.
    So the LCM of 12, 15, and 20 is 60.

Why this feels “shortest”

  • No long lists of multiples.
  • Works for 3, 4, or more numbers by the same idea.
  • Very systematic, so it’s easy to avoid mistakes.

Method 2: Ladder / Division “Shortcut”

Many teachers call this the ladder method or prime division method —it’s like a visual shortcut built from prime factorization.

Steps (example with 6, 8, 10)

  1. Write the three numbers in a row.
    Start with: 6 8 10
  1. Divide by a prime that divides at least two numbers.
    • Choose 2 (since it divides all three).
    • Divide:
      • 6 ÷ 2 = 3
      • 8 ÷ 2 = 4
      • 10 ÷ 2 = 5
    • New row: 3 4 5
  1. Repeat until no prime divides at least two numbers.
    In 3, 4, 5 there is no prime that divides at least two of them (only 1 is common), so you stop.
  1. Multiply all numbers outside the ladder.
    • Outside primes: 2
    • Inside “final” row: 3, 4, 5
    • LCM =2×3×4×5=120=2\times 3\times 4\times 5=120=2×3×4×5=120.

This gives the same result as prime factorization but in a compact, visual way, which many learners find quicker in practice.

Method 3: Two‑Step LCM (Using Pair‑wise LCM)

Another efficient trick is: find LCM of two numbers, then combine with the third.

Steps (example: 4, 6, 10)

  1. Find LCM of first two: LCM(4, 6) = 12.
  1. Then find LCM of that result with the third: LCM(12, 10).
    • Prime factorization:
      • 12 = 22×32^2\times 322×3
      • 10 = 2×52\times 52×5
      • LCM = 22×3×5=602^2\times 3\times 5=6022×3×5=60.

So LCM(4, 6, 10) = 60. This is especially handy if you already know or can quickly spot the LCM of a pair.

Which Way Is Actually “Shortest”?

Here’s a compact view of the main methods mentioned in recent learning resources:

[8][2] [9][3][5] [6][5] [5][6]
Method Idea Best for Why it feels shortest
Prime factorization Break each number into primes, take highest powers, multiply. Medium to large numbers, exam work. Few, clear steps; very systematic.
Ladder / division method Keep dividing the row of numbers by primes, then multiply outside and remaining numbers. Classroom practice, visual learners. Looks like a quick “trick”; avoids writing factor trees.
Pair‑wise LCM using GCF/LCM Find LCM of first two, then LCM of result with third. When two numbers are easy to combine first. Reduces three‑number problem into two simple ones.
Listing multiples Write multiples until a common one appears. Very small numbers only. Conceptually simple, but can get long.
For most three‑number LCM questions, teachers and learning sites highlight prime factorization or the ladder/division method as the **fastest** approaches.

Mini Example Story (to Remember It)

Imagine three runners:

  • Runner A laps a 200 m track every 12 minutes.
  • Runner B every 15 minutes.
  • Runner C every 20 minutes.

You want to know when they will all hit the starting line together again. You’re really asking for the LCM of 12, 15, and 20 , which you saw is 60 minutes.

Prime factorization or the ladder method gets you that answer without having to write out “12, 24, 36, …” and “15, 30, 45, …” and “20, 40, 60, …” over and over.

TL;DR

  • The shortest way for three numbers is usually prime factorization or the ladder/division method.
  • Prime factorization: factor each, take highest powers of each prime, multiply.
  • Ladder method: divide the row of numbers by primes step by step, then multiply the divisors and the last row.

Bottom note: Information gathered from public forums or data available on the internet and portrayed here.