The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digi… — State PSC Exam

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digit integers $P$ and $Q$ (where $P > Q$ ) are $24$ and $1080$ , respectively. If the difference between the two numbers is $96$ , what is the sum of the two numbers ( $P + Q$ )?

Answer: 336

Answer

The sum of the two numbers P and Q is 336.

By representing

P = 24a

and

Q = 24b

with

\gcd(a, b) = 1

, the relation

\text{HCF} \times \text{LCM} = P \times Q

yields

a \times b = 45

. The coprime factor pairs of

45

are

(45, 1)

and

(9, 5)

. The pair

(45, 1)

gives

1080

and

24

, which are not both three-digit numbers. The pair

(9, 5)

gives

P = 216

and

Q = 120

, both of which are three-digit numbers with a difference of

96

. The sum of these two numbers is

216 + 120 = 336

Step-by-Step Solution

Express the two numbers in terms of their HCF and coprime factors

Let

P = 24a

and

Q = 24b

where

\gcd(a, b) = 1

and

a > b

Any two numbers sharing an HCF of

h

can be represented as

h \cdot a

and

h \cdot b

where

a

and

b

have no common prime factors.

Relate the product of the coprime factors to the LCM and HCF

a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{1080}{24} = 45

The product of two numbers equals the product of their HCF and LCM:

(24a)(24b) = 24 \times 1080

Identify all coprime factor pairs of 45

The coprime factor pairs

(a, b)

with

a > b

are

(45, 1)

and

(9, 5)

Factor pairs such as

(15, 3)

are invalid because

\gcd(15, 3) = 3 \neq 1

Apply the three-digit integer and difference constraints to select the valid pair

For

(a, b) = (9, 5)

P = 24 \times 9 = 216

and

Q = 24 \times 5 = 120

. Difference =

216 - 120 = 96

The pair

(45, 1)

yields

P = 1080

(four digits) and

Q = 24

(two digits), failing the three-digit criteria.

Calculate the sum of

P

and

Q

P + Q = 216 + 120 = 336

The question asks specifically for the sum

P + Q

Key Concept

Properties of HCF and LCM including HCF x LCM = Product of Numbers and Prime Factor Coprimality

Question

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