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HCF and LCM

Competitive Mathematics · Number System

8 Part1 worked examples4 practice questions

HCF (highest common factor) and LCM (lowest common multiple) are among the most-used tools in number work. When the question says “the greatest measure that divides all exactly” you need the HCF; when it says “after how long will they all coincide again” you need the LCM. This chapter covers both methods, how they are related, and every type of remainder, bell and tile question.

  1. Understand the idea — what HCF and LCM really are.
  2. Study the solved examples — ask “why?” at every step.
  3. Try it yourself — solve on paper first, then choose.
1

Factors and multiples

The two foundation ideas

What factors and multiples are, and how HCF and LCM come from shared factors and shared multiples.

Factors, multiples and what they share

Down and up

Numbers that cut a number exactly are its factors; the bigger numbers it builds are its multiples.

The numbers 1, 2, 3, 4, 6 and 12 divide 12 exactly — they are its factors. The multiples of 12 go the other way: 12, 24, 36, 48, … Factors are never bigger than the number; multiples are never smaller.

Of 12Of 18Common
Factors1, 2, 3, 4, 6, 121, 2, 3, 6, 9, 181, 2, 3, 6
Multiples12, 24, 36, 48, 60, 7218, 36, 54, 72, 9036, 72, …
  • HCF = the greatest of the common factors (here 6).
  • LCM = the least of the common multiples (here 36).
  • The HCF is never bigger than the numbers; the LCM is never smaller.
TrickAn easy memory: HCF = the Highest shared divisor (the one that cuts). LCM = the Lowest shared multiple (the one that is built).
Example 1 · Basic
Find the HCF and LCM of 12 and 18 by listing.
Solution, step by step
  1. 1Factors of 12: 1, 2, 3, 4, 6, 12; of 18: 1, 2, 3, 6, 9, 18.
  2. 2Common: 1, 2, 3, 6 → the largest is 6. HCF = 6.
  3. 3Multiples of 12: 12, 24, 36, …; of 18: 18, 36, … → the first shared is 36. LCM = 36.
Answer HCF = 6, LCM = 36
BasicTry it yourself first
Find the HCF of 24, 36.
BasicTry it yourself first
Find the LCM of 8, 12.
BasicTry it yourself first
Find the HCF of 30, 45.
BasicTry it yourself first
Find the LCM of 6, 15.
2

Methods for the HCF

Factorisation and division

Finding the HCF of two or three numbers using prime factors and the long-division (Euclid) method.

The prime-factor method

Shared primes at the smallest power

Break each number into primes and multiply the shared primes at their smallest power.

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The long-division (Euclid) method

For big numbers

When the numbers are big and factorising is hard, find the HCF by repeated division.

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3

Methods for the LCM

Factorisation and the ladder

Finding the LCM using prime factors and the common-division (ladder) method.

The prime-factor method

Every prime at its biggest power

Take every prime that appears and multiply each at its biggest power.

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The ladder (common-division) method

Divide all the numbers together

Divide by a prime, cutting every number it divides, and build a ladder.

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4

The relation and fractions

Product = HCF × LCM

The link between the product of two numbers and their HCF and LCM, and the HCF and LCM of fractions.

The product of two numbers

The most-asked formula

and how to use it.

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HCF and LCM of fractions

Numerators and denominators separately

The HCF and LCM of fractions (and decimals), by working on numerators and denominators separately.

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5

Applications of HCF and LCM

Remainders · bells · tiles · ratios

Every type of word problem in exams, and how to choose between HCF and LCM in each.

HCF or LCM — how to choose

From the wording

In every word problem ask one question: am I splitting, or repeating?

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Remainder questions

Three types

The remainder is the same, or different with an equal shortfall, or a least number is wanted.

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Bells, lights and laps

When together again?

When several things repeat at different intervals, the time they coincide.

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Equal pieces: tiles, ropes, groups

The greatest measure = HCF

When several lengths or counts must be split into equal parts and the part must be as large as possible.

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n-digit numbers

The greatest and the smallest

The greatest or smallest number with a given number of digits that is divisible by several divisors.

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Ratios and the HCF

Ratio numbers × HCF

When the ratio and the HCF of two numbers are given: the actual numbers, their sum or LCM.

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6

Choosing the method, and the traps

Clues · traps

Picking HCF or LCM from the wording, and the commonest slips.

The commonest mistakes

Six traps

In HCF-LCM marks are mostly lost to choosing the wrong method.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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