An inventory analyst is auditing two specific electronic components in a warehouse. He cal… — State PSC Exam

An inventory analyst is auditing two specific electronic components in a warehouse. He calculates that the product of their exact unit quantities is $32,175$ , and the Highest Common Factor (HCF) of these two quantities is $15$ . If there are more than $100$ units of each component currently in stock, what is the total combined quantity of both components?

Cevap: 360 units

Cevap

The correct total combined quantity is 360 units.

By representing the numbers as

15x

and

15y

, we determine that

15x \times 15y = 32,175

, which simplifies to

xy = 143

. Factoring

143

into co-prime pairs gives

(1, 143)

and

(11, 13)

. Multiplying these by the HCF (

15

) gives potential quantities of

(15, 2145)

and

(165, 195)

. Only the pair

(165, 195)

satisfies the condition that both quantities must be greater than

100

. Their sum is

165 + 195 = 360

Adım Adım Çözüm

Express the two unknown quantities mathematically using their HCF.

Let the quantities be

15x

and

15y

, where

x

and

y

are co-prime integers.

Since the HCF of the two numbers is

15

, both numbers must be multiples of

15

. Their remaining factors (

x

and

y

) cannot share any common factors other than

1

Formulate an equation using the given product of the quantities.

15x \times 15y = 32,175

The problem states the product of the two component quantities is equal to

32,175

Solve the equation for the product of the co-prime variables

x

and

y

225xy = 32,175 \Rightarrow xy = \frac{32,175}{225} = 143

Isolating

xy

simplifies the problem to finding two co-prime factors that multiply to

143

Identify all co-prime factor pairs of

143

The co-prime pairs are

(1, 143)

and

(11, 13)

We must list all integer pairs that multiply to

143

and verify they share no common divisors.

Calculate the possible original quantities and apply the boundary constraints.

Pair 1 gives

(15, 2145)

. Pair 2 gives

(165, 195)

. Because the problem states both quantities are

>100

, we must choose

(165, 195)

The constraint strictly eliminates the first pair, as

15

is not greater than

100

Sum the valid quantities.

165 + 195 = 360

The question asks for the total combined quantity of both components.

Anahtar Kavram

The relationship between the Highest Common Factor (HCF) and the product of two numbers, utilizing co-prime factor pairs.

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