HCF & LCM

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Subtopic 1 of 4·Finding HCF & LCM (incl. fractions and decimals) →

HCF & LCM: definitions and core relations

HCF (GCD) of some numbers is the largest number that divides all of them. LCM is the smallest number divisible by all of them.

HCF ≤ every number ≤ LCM (for numbers ≥ 1).
LCM is always a multiple of the HCF.
For co-prime numbers: HCF = 1 and LCM = product.
For two numbers: $\text{HCF} \times \text{LCM} =$ product of the numbers. (Does not hold for three numbers.)
If one number and the HCF are given, the other number $= \frac{\text{HCF} \times \text{LCM}}{\text{given number}}$ .

HCF of numbers close together is small (it must divide their difference); HCF of numbers in a fixed ratio is the scale factor.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 15 practice Q

Detailed notes

What are HCF and LCM?

HCF (Highest Common Factor, also called GCD) of some numbers is the largest number that divides each of them exactly. LCM (Lowest Common Multiple) is the smallest number that is exactly divisible by each of them. Example: for 12 and 18 the common factors are 1, 2, 3, 6, so HCF $= 6$ ; the multiples of 12 and 18 first meet at 36, so LCM $= 36$ .

Five rules that solve most questions

Size order: HCF $\le$ every number $\le$ LCM (for numbers $\ge 1$ ). An option that breaks this is wrong at sight.
Product relation (two numbers only): $\text{HCF} \times \text{LCM} = a \times b$ . Check: $6 \times 36 = 216 = 12 \times 18$ . It does not hold for three numbers — never use it there.
Missing number: if one number, the HCF and the LCM are known, the other number $= \frac{\text{HCF} \times \text{LCM}}{\text{given number}}$ . Example: HCF 18, LCM 378, one number 54 → other $= \frac{18 \times 378}{54} = 126$ .
Co-prime numbers (numbers whose HCF is 1): LCM = product. Consecutive integers are always co-prime, so their LCM is just their product.
LCM is always a multiple of the HCF. An option that is not a multiple of the given HCF is wrong at sight.

Numbers given in a ratio

Numbers in ratio $a : b$ (with a, b co-prime) and HCF $h$ are $ha$ and $hb$ :

LCM $= h \cdot a \cdot b$ , sum $= h(a + b)$ , difference $= h(b - a)$ , product $= h^2 ab$ . Example: ratio 3 : 4, HCF 6 → numbers 18 and 24 → LCM $= 6 \times 3 \times 4 = 72$ , sum $= 42$ . Working backwards: if a 5 : 7 pair has LCM 140, then $h \times 5 \times 7 = 140$ → $h = 4$ → numbers 20 and 28.

HCF of numbers that are close or related

The HCF divides every difference: $\gcd(a, b)$ divides $a - b$ . So two numbers 20 apart can never have HCF 12 — a fast option-killer.
Consecutive integers: HCF = 1, LCM = their product.
"Other factors of the LCM": if the HCF is $h$ and the remaining factors of the LCM are $m$ and $n$ (co-prime), the numbers are $hm$ and $hn$ , and LCM $= hmn$ . Example: HCF 23 with other factors 13 and 14 → numbers 299 and 322, LCM $= 23 \times 13 \times 14 = 4186$ .

Counting pairs (a preview)

"How many pairs of numbers have product P and HCF h?" — write them as $ha, hb$ with $\gcd(a, b) = 1$ , so $ab = \frac{P}{h^2}$ . The answer is the number of co-prime factor pairs of $\frac{P}{h^2}$ (worked fully in the two-step-cases subtopic).

Quick revision

Two numbers: HCF × LCM = product. Three numbers: no such rule.
Co-prime → HCF 1, LCM = product. Consecutive integers are co-prime.
Ratio a : b with HCF h → numbers ha, hb; LCM = hab; sum = h(a + b).
LCM is a multiple of the HCF; HCF divides every difference.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Find the HCF or LCM of small numbers directlyvery common2 practice Q

How to spot it:

'Find the HCF of 96 and 404' or 'the LCM of 12, 15 and 20 is' — plain computation, no story.

\text{HCF} = \text{common primes, lowest powers}; \quad \text{LCM} = \text{all primes, highest powers}

Factorise each number into primes.
HCF: multiply the primes common to ALL numbers, taking the lowest power of each.
LCM: multiply every prime that appears, taking the highest power of each.

Why: a common divisor cannot use more of any prime than the number with the fewest; a common multiple needs at least the richest stock of every prime.

Example: Find the LCM of 16, 24 and 36.

$16 = 2^4$ , $24 = 2^3 \cdot 3$ , $36 = 2^2 \cdot 3^2$ . LCM $= 2^4 \times 3^2 = 144$ .

Type 2: HCF × LCM = product (two numbers)very common2 practice Q

How to spot it:

Any two of {product, HCF, LCM, one number} are given and a third is asked.

\text{HCF} \times \text{LCM} = a \times b \ \Rightarrow\ b = \frac{\text{HCF} \times \text{LCM}}{a}

Write the product relation (valid for exactly two numbers).
Fill in the three known values and divide to get the fourth.
Sanity check: the HCF must divide the LCM.

Why: writing $a = ha'$ , $b = hb'$ with co-prime $a', b'$ , the LCM is $ha'b'$ , so HCF × LCM $= h^2 a'b' = ab$ .

Example: The HCF of two numbers is 18 and their LCM is 378. If one number is 54, find the other.

Other $= \frac{18 \times 378}{54} = \frac{6804}{54} = 126$ . Check: $\gcd(54, 126) = 18$ and LCM $= 378$ .

Type 3: Numbers in a given ratio, HCF or LCM knownvery common2 practice Q

How to spot it:

'Two numbers are in ratio m : n and their HCF (or LCM) is …' — then a sum, difference or the LCM is asked.

\text{numbers} = hm, hn; \quad \text{LCM} = hmn, \quad \text{sum} = h(m+n)

Treat the ratio parts m, n as co-prime (cancel the ratio first if needed).
HCF given: numbers are hm and hn. LCM given: $h = \frac{\text{LCM}}{mn}$ .
Answer whatever is asked (sum, difference, product, LCM).

Why: the ratio fixes only the shape; the HCF is the scale factor h.

Example: Two numbers are in the ratio 5 : 7 and their LCM is 140. Find the difference of the numbers.

$h = \frac{140}{5 \times 7} = 4$ → numbers 20 and 28 → difference $8$ .

Type 4: HCF given with the 'other factors' of the LCMcommon2 practice Q

How to spot it:

'The HCF of two numbers is 29 and the other two factors of their LCM are 15 and 13. Find the numbers.'

\text{numbers} = hm, hn \ \ (m, n \text{ co-prime}); \quad \text{LCM} = hmn

The numbers are h × (first other factor) and h × (second other factor).
The two 'other factors' must be co-prime — if not, the data is impossible.
LCM = h × m × n if it is asked too.

Why: LCM = HCF × (leftover co-prime parts), by the structure of HCF and LCM.

Example: The HCF of two numbers is 23 and the other two factors of their LCM are 13 and 14. Find the greater number.

Numbers: $23 \times 13 = 299$ and $23 \times 14 = 322$ . Greater $= 322$ . (LCM would be $23 \times 13 \times 14 = 4186$ .)

Type 5: Co-prime, consecutive and other property questionscommon2 practice Q

How to spot it:

'The LCM of two co-prime numbers is …, one is …' or 'which pair is co-prime?' — the word co-prime or consecutive appears.

\gcd(a, b) = 1 \Rightarrow \text{LCM} = ab; \qquad \gcd(a, b) \mid (a - b)

Co-prime + LCM given: divide the LCM by the given number to get the other.
'Which pair is co-prime?': test each pair for a common factor > 1.
Difference-based checks: the HCF must divide the difference of the numbers.

Why: with no common prime, nothing cancels — so the LCM is the plain product.

Example: The LCM of two co-prime numbers is 221 and one of them is 13. Find the other.

Co-prime → LCM = product → other $= \frac{221}{13} = 17$ . (Indeed $221 = 13 \times 17$ .)

Formulas

Product relation (two numbers)

\text{HCF} \times \text{LCM} = a \times b

Other number

b = \frac{\text{HCF} \times \text{LCM}}{a}

Co-prime case

\gcd(a,b) = 1 \Rightarrow \text{LCM}(a,b) = ab

HCF divides every difference

\gcd(a,b) \mid (a-b)

LCM multiple of HCF

\gcd(a,b) \mid \text{LCM}(a,b)

Shortcut tricks

⚡ Ratio split

Numbers in ratio $a : b$ (a, b co-prime) with HCF $h$ are $ha$ and $hb$ ; LCM $= hab$ ; sum $= h(a+b)$ ; difference $= h(b-a)$ .

Example: Two numbers are in ratio 5 : 6 and their HCF is 12. Find the LCM.

Numbers are 60 and 72. LCM $= 12 \times 5 \times 6 = 360$ .

⚡ Product ÷ HCF = LCM in one step

Any question giving two of {product, HCF, LCM, one number} resolves by the product relation.

Example: The product of two numbers is 1600 and their HCF is 5. Find the LCM.

$\text{LCM} = \frac{1600}{5} = 320$ .

⚡ Divisor pairs of (product ÷ HCF)

For 'how many pairs of numbers' questions, split product/HCF into co-prime factor pairs.

Example: The product of two numbers is 2028 and HCF is 13. How many such pairs exist?

$\frac{2028}{13} = 156 = 13a \times 13b / 13^2$ ... i.e. $a \times b = 156$ with $\gcd(a,b)=1$ : pairs (1,156), (3,52), (4,39), (12,13) ⇒ 4 pairs.

Where students lose marks

Using $\text{HCF} \times \text{LCM} = abc$ for three numbers — false for three.
Forgetting that the co-prime pair (1, product) counts as a valid pair.
Confusing 'greatest number that divides' (HCF) with 'least number that is divisible by' (LCM).
Assuming the HCF of a ratio pair is the ratio itself.

Practice sets — 15 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.