AFCAT Numerical Ability · LCM and HCF

Finding the HCF and LCM

Write each number as a product of primes, then take the lowest common powers for the HCF and the highest powers for the LCM; use repeated division for large numbers, and a numerator and denominator rule for fractions.

All LCM and HCF notes

Why this matters

These questions hand you the numbers and ask for the HCF or the LCM, so the only marks lost are in the factorising. The usual slip is mixing the two rules: taking a prime into the HCF that only one number has, or dropping a higher power from the LCM. Fractions add one more rule that is easy to turn upside down.

Concept 1 of 3: HCF and LCM by prime factors

The HCF (highest common factor) is the largest number that divides every given number. The LCM (least common multiple) is the smallest number that every given number divides. Once each number is written as powers of primes, both can be read off: the HCF keeps only what all the numbers share, and the LCM keeps everything any of them has.

Definition

Factorise each number into prime powers.
HCF: take each prime that appears in every number, at its lowest power.
LCM: take every prime that appears in any number, at its highest power.
Two numbers with no common prime (coprime) have HCF 1, and their LCM is their product.
For two numbers, HCF × LCM equals the product of the numbers. The next page builds on this.

HCF and LCM from prime powers

\text{HCF} = \prod p^{\min(a, b)}, \qquad \text{LCM} = \prod p^{\max(a, b)}

Worked example

Find the HCF and the LCM of 72 and 120.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · LCM and HCF · Finding HCF and LCM

The HCF of 1056, 1584 and 2178 is

The HCF keeps only shared primes

For

2^3 \times 5

and

2 \times 3

, the HCF is 2. The 5 and the 3 belong to one number each, so they go into the LCM (which is 120) and never into the HCF.

The LCM takes the highest power

The LCM of

2^2 \times 3

and

2 \times 3^2

2^2 \times 3^2 = 36

. Taking

2 \times 3

from either number gives the HCF, 6, by mistake.

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Concept 2 of 3: HCF of large numbers by repeated division

Large numbers such as 667 are slow to factorise. Any number that divides two numbers also divides their difference, and so it divides the remainder when the larger is divided by the smaller. Keep dividing the last divisor by the last remainder until nothing is left over.

Definition

Divide the larger number by the smaller.
Divide the divisor by the remainder. Repeat.
When the remainder is 0, the last divisor is the HCF.
For three numbers, find the HCF of two, then the HCF of that answer and the third.

Euclid's division step

\text{HCF}(a, b) = \text{HCF}(b,\; a \bmod b)

Worked example

Find the HCF of 391 and 667.

Practice this conceptself-check · 3 quick reps

Divide by the remainder

Each step divides the last divisor by the last remainder. Dividing the first number again by the new remainder breaks the chain and gives a wrong HCF.

The answer is the last divisor

When the remainder reaches 0, the HCF is the number you divided by in that step, not the quotient and not the remainder before it.

Concept 3 of 3: HCF and LCM of fractions

The HCF of fractions is the largest fraction that goes into each of them a whole number of times; the LCM is the smallest fraction that each of them goes into a whole number of times. The numerators behave like ordinary numbers, and the denominators do the opposite job.

Definition

Write every fraction in its lowest terms first.
HCF of fractions = HCF of the numerators ÷ LCM of the denominators.
LCM of fractions = LCM of the numerators ÷ HCF of the denominators.
Decimals: give them the same number of decimal places, work with the whole numbers, then put the point back. The HCF of 1.2 and 0.84 is the HCF of 120 and 84 hundredths, 0.12.

Fractions

\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}

Worked example

Find the HCF and the LCM of

\tfrac 23

\tfrac 49

and

\tfrac{8}{15}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · LCM and HCF · Finding HCF and LCM

The HCF of

\frac{5}{6}

\frac{10}{18}

and

\frac{25}{36}

The denominators swap jobs

For the HCF of fractions the denominators take their LCM, not their HCF. Check any answer by dividing: each fraction divided by the HCF must give a whole number.

Lowest terms first

\tfrac 24

and

\tfrac 36

are both

\tfrac 12

, so their HCF is

\tfrac 12

. Applying the rule to the unreduced fractions gives

\tfrac{1}{12}

, which is wrong.

Drill 1 more on hcf and lcm of fractions

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 (3)

HCF and LCM by prime factors
HCF and LCM from prime powers
$\text{HCF} = \prod p^{\min(a, b)}, \qquad \text{LCM} = \prod p^{\max(a, b)}$
HCF of large numbers by repeated division
Euclid's division step
$\text{HCF}(a, b) = \text{HCF}(b,\; a \bmod b)$
HCF and LCM of fractions
Fractions
$\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$

Watch out for (6)

The HCF keeps only shared primes
→ HCF and LCM by prime factors
The LCM takes the highest power
→ HCF and LCM by prime factors
Divide by the remainder
→ HCF of large numbers by repeated division
The answer is the last divisor
→ HCF of large numbers by repeated division
The denominators swap jobs
→ HCF and LCM of fractions
Lowest terms first
→ HCF and LCM of fractions

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