AFCAT Numerical Ability · Number System

Factors and Multiples: HCF, LCM and Co-prime Numbers

List factors and multiples, find the HCF (the greatest common factor) and the LCM (the least common multiple), and use HCF × LCM = product for two numbers.

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Why this matters

None of this page is HARD, but HCF and LCM are easy to swap. Every question asks either for the greatest number that divides (the HCF) or the smallest number that is divided (the LCM), and settling which one comes before any arithmetic.

Concept 1 of 4: Factors, multiples and the common ones

A factor of

n

divides it exactly; a multiple of

n

n

times a whole number. Factors come in pairs that multiply to

n

, so you only need to search up to

\sqrt n

. Multiples never stop.

Definition

List the factors of $n$ in pairs: for 28, $1 \times 28$ , $2 \times 14$ , $4 \times 7$ .
Every number has 1 and itself as factors. Factors are at most $n$ ; multiples are at least $n$ .
Common factors of two numbers appear in both lists. They are exactly the factors of the HCF (next concept).
Common multiples are divisible by both. They are exactly the multiples of the LCM.

Factor pairs

n = a \times b \;\Rightarrow\; a \text{ and } b \text{ are both factors of } n

Worked example

List the factors of 42 and of 56, and their common factors.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Number System · Factors and Multiples

Find the common factors of

30

and

45

Factors versus multiples

The factors of 24 run from 1 to 24; its multiples start at 24 and go on (48, 72, ...). An option list of multiples offered as factors is the wrong answer.

Do not drop 1 or the number itself

A factor list of 28 without 1 or 28 is incomplete, and so is a list of common factors without 1.

Drill 4 more on factors, multiples and the common ones

Concept 2 of 4: HCF, by primes or by Euclid, and co-prime numbers

The highest common factor is the largest number that divides each of them. From the prime factorisations, it keeps only the primes every number shares, each to the smallest power found. For two large numbers, Euclid's method replaces the larger by its remainder until the division comes out exact.

Definition

By primes: common primes, each to its lowest power.
By Euclid: divide the larger by the smaller, then divide the smaller by the remainder, and so on. The last non-zero remainder is the HCF.
'The greatest number that divides $a$ , $b$ and $c$ exactly' is their HCF.
The number of common factors is the number of divisors of the HCF.
Co-prime numbers have HCF 1. Neither need be prime: 8 and 15 are co-prime.

HCF from prime factors

\text{HCF} = \text{product of the common primes, each to its lowest power}

Worked example

Find the HCF of 84 and 126, by primes and by Euclid.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Number System · Factors and Multiples

Find the greatest number that divides

72

96

, and

120

Co-prime is not prime

8 and 15 are co-prime though neither is prime. Two odd numbers need not be co-prime: 15 and 21 share the factor 3.

Lowest powers for HCF

For

2^3 \times 5

and

2 \times 5^2

the HCF is

2 \times 5 = 10

. Taking the highest powers gives the LCM instead.

Drill 6 more on hcf, by primes or by euclid, and co-prime numbers

Concept 3 of 4: LCM, and the least number leaving a given remainder

The least common multiple is the smallest number that each of them divides. It must hold every prime any of them needs, each to the largest power any of them needs, and nothing more. Lights that flash together again, or bells that ring together, meet after the LCM.

Definition

By primes: every prime that appears, each to its highest power.
'The smallest number divisible by $a$ , $b$ and $c$ ', 'when will they ring (flash, meet) together' are LCM questions.
The least number that leaves the same remainder $r$ on division by $a$ , $b$ and $c$ is $\text{LCM} + r$ .
For numbers that share no factor, the LCM is their product.

LCM from prime factors

\text{LCM} = \text{product of every prime present, each to its highest power}

Worked example

Three lights flash every 8, 14 and 18 seconds. They flash together now. After how many seconds will they next flash together?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Number System · Factors and Multiples

Four bells ring at intervals of

6

sec,

12

sec,

15

sec, and

20

sec. After how many seconds will they ring together?

The LCM is not always the product

8 and 14 share the factor 2, so their LCM is 56, not

8 \times 14 = 112

. The product is the LCM only for co-prime numbers.

Add the remainder, do not subtract it

A number leaving remainder 3 on 4, 9 and 14 is 3 more than a common multiple: 255, not 249.

Drill 5 more on lcm, and the least number leaving a given remainder

Concept 4 of 4: HCF × LCM = product of the two numbers

For two numbers, the HCF collects the primes they share and the LCM collects everything; between them they use each prime exactly as often as the two numbers do. So the HCF times the LCM is the product of the numbers.

Definition

For two numbers $a$ and $b$ : $\text{HCF} \times \text{LCM} = a \times b$ .
Given the HCF, the LCM and one number, the other is $\tfrac{\text{HCF} \times \text{LCM}}{a}$ .
The HCF always divides the LCM. If it does not, no such pair exists.
For co-prime numbers the HCF is 1, so the LCM is the product.

Two numbers

\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b

Worked example

The HCF of two numbers is 12 and their LCM is 360. One number is 72. Find the other.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Number System · Factors and Multiples

The LCM of two numbers is

210

and their HCF is

14

. If one number is

42

, find the other.

Two numbers only

For 2, 4 and 8 the HCF is 2 and the LCM is 8, so HCF × LCM is 16, but the product is 64. The rule holds for a pair, not for three numbers.

Drill 2 more on hcf × lcm = product of the two numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Factors, multiples and the common ones
Factor pairs
$n = a \times b \;\Rightarrow\; a \text{ and } b \text{ are both factors of } n$
HCF, by primes or by Euclid, and co-prime numbers
HCF from prime factors
$\text{HCF} = \text{product of the common primes, each to its lowest power}$
LCM, and the least number leaving a given remainder
LCM from prime factors
$\text{LCM} = \text{product of every prime present, each to its highest power}$
HCF × LCM = product of the two numbers
Two numbers
$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Watch out for (7)

Factors versus multiples
→ Factors, multiples and the common ones
Do not drop 1 or the number itself
→ Factors, multiples and the common ones
Co-prime is not prime
→ HCF, by primes or by Euclid, and co-prime numbers
Lowest powers for HCF
→ HCF, by primes or by Euclid, and co-prime numbers
The LCM is not always the product
→ LCM, and the least number leaving a given remainder
Add the remainder, do not subtract it
→ LCM, and the least number leaving a given remainder
Two numbers only
→ HCF × LCM = product of the two numbers

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