AFCAT Numerical Ability · LCM and HCF

HCF and LCM Together: the Product Rule and Coprime Parts

Two numbers multiply to their HCF times their LCM, and each can be written as the HCF times a coprime part; between them these two facts find a missing number from the HCF, the LCM, a ratio or a sum.

All LCM and HCF notes

Why this matters

AFCAT asks this more than any other part of the chapter: three of its four past-year questions are here, from 2015 and 2018. Each gives two of the HCF, the LCM, one number, a ratio or a sum and asks for the rest. Marks go when the product rule is used for three numbers, or when a ratio is not first written in its lowest terms.

Concept 1 of 3: Product of two numbers equals HCF times LCM

Every prime of two numbers lands once in the HCF at its lower power and once in the LCM at its higher power. Together they hold exactly the primes of both numbers, so HCF × LCM is the product of the two numbers.

Definition

For two numbers $a$ and $b$ : $a \times b = \text{HCF} \times \text{LCM}$ .
So the other number is $\dfrac{\text{HCF} \times \text{LCM}}{\text{one number}}$ .
If the LCM is a multiple of the HCF, say $L = mH$ , and their sum or difference is given, find $H$ first.
If the product of HCF and LCM is given, it is the product of the numbers: pair it with the sum or difference.

Product rule (two numbers only)

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

Worked example

The HCF of two numbers is 6 and their LCM is 180. One number is 36. Find the other.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (I) 2018 — Numerical Ability · 01-402Easy

Example 1 · LCM and HCF · HCF and LCM Together

The LCM of two numbers is 520 and their HCF is 4. If one of the numbers is 52, then the other number is

Two numbers only

For 3, 6 and 12 the product is 216, but HCF × LCM is

3 \times 12 = 36

. The product rule holds for a pair of numbers, never for three or more.

Check the answer is a whole number

If HCF × LCM divided by the given number is not a whole number, or its HCF with that number is not the HCF given, recheck the arithmetic before picking an option.

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Concept 2 of 3: Numbers as the HCF times coprime parts

If the HCF of two numbers is

H

, they are

Ha

and

Hb

, where

a

and

b

share no factor. Then the LCM is

Hab

. A ratio in lowest terms gives

a

and

b

directly, and the HCF is the multiplier.

Definition

Numbers in the ratio $a : b$ (lowest terms) are $ak$ and $bk$ , with HCF $k$ and LCM $abk$ .
Three numbers $ak, bk, ck$ : the HCF is $k$ when $a, b, c$ have no common factor, and the LCM is $\text{LCM}(a, b, c) \times k$ .
The HCF divides the sum and the difference of the numbers. Given the sum and the LCM, the HCF is the HCF of those two, because $H(a + b)$ and $Hab$ share only $H$ .
If both numbers are multiples of a given number, write them as that number times a part and work with the parts.

Coprime parts

a' = Ha,\; b' = Hb,\; \text{HCF}(a, b) = 1 \;\Rightarrow\; \text{LCM} = Hab

Worked example

Two numbers are in the ratio

3 : 5

and their LCM is 225. Find the numbers and their HCF.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 01-404Moderate

Example 2 · LCM and HCF · HCF and LCM Together

Find two natural numbers whose sum is 85 and the least common multiple is 102.

Reduce the ratio first

Numbers in the ratio

6 : 8

with HCF 5 are 15 and 20, because

6 : 8 = 3 : 4

. Writing 30 and 40 gives a pair whose HCF is 10.

The LCM of the ratio terms, not their product

Numbers

3k, 4k, 6k

have LCM

12k

, since

\text{LCM}(3, 4, 6) = 12

. Multiplying the terms to get

72k

works only when the terms share no factor.

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Concept 3 of 3: What the HCF and LCM can and cannot be

Some questions give one of the two and ask which value the other could be, or could never be. The facts below settle them without finding any numbers.

Definition

The HCF divides each number and each number divides the LCM, so the HCF always divides the LCM. A value that fails one of these facts is impossible.

Fact	Example
The LCM is a multiple of the HCF	HCF 6 allows LCM 90, since $90 = 15 \times 6$
HCF ≤ the smaller number, LCM ≥ the larger	14 and 35: HCF 7, LCM 70
Coprime numbers: HCF 1, LCM is the product	9 and 16: HCF 1, LCM 144
One number divides the other: HCF is the smaller, LCM the larger	13 and 39: HCF 13, LCM 39
Two consecutive numbers are coprime	20 and 21: HCF 1, LCM 420

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · LCM and HCF · HCF and LCM Together

The HCF of two numbers is 8. Which one of the following can never be their LCM?

Ruling out is not proving

Any divisor of the LCM could be the HCF. A question asking what the HCF will be can only rule out the options that do not divide the LCM; the one left is the answer only because the others fail.

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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 (2)

Product of two numbers equals HCF times LCM
Product rule (two numbers only)
$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$
Numbers as the HCF times coprime parts
Coprime parts
$a' = Ha,\; b' = Hb,\; \text{HCF}(a, b) = 1 \;\Rightarrow\; \text{LCM} = Hab$

Reference tables (1)

What the HCF and LCM can and cannot be5 rows

Fact	Example
The LCM is a multiple of the HCF	HCF 6 allows LCM 90, since $90 = 15 \times 6$
HCF ≤ the smaller number, LCM ≥ the larger	14 and 35: HCF 7, LCM 70
Coprime numbers: HCF 1, LCM is the product	9 and 16: HCF 1, LCM 144
One number divides the other: HCF is the smaller, LCM the larger	13 and 39: HCF 13, LCM 39
Two consecutive numbers are coprime	20 and 21: HCF 1, LCM 420

Watch out for (5)

Two numbers only
→ Product of two numbers equals HCF times LCM
Check the answer is a whole number
→ Product of two numbers equals HCF times LCM
Reduce the ratio first
→ Numbers as the HCF times coprime parts
The LCM of the ratio terms, not their product
→ Numbers as the HCF times coprime parts
Ruling out is not proving
→ What the HCF and LCM can and cannot be

Drill the questions on this topic

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