Two positive integers A and B (both strictly greater than 12) have a Highest Common Factor… — State PSC Exam

Two positive integers $A$ and $B$ (both strictly greater than $12$ ) have a Highest Common Factor (HCF) of $12$ and a Least Common Multiple (LCM) of $504$ . Based on this information, which of the following statements are true?

The sum of the integers $A$ and $B$ can be $156$ .Answer
The positive difference between $A$ and $B$ can be $132$ .Answer
C
There are exactly four distinct unordered pairs $\{A, B\}$ that satisfy all given conditions.
D
The Least Common Multiple (LCM) of the fractions $\frac{1}{A}$ and $\frac{1}{B}$ is $\frac{1}{504}$ .

Answer

The statements confirming that the sum of the integers can be 156 and that their positive difference can be 132 are both correct.

Based on the prime factorization constraints, the valid unordered pairs

\{A, B\}

are

\{24, 252\}

\{36, 168\}

, and

\{72, 84\}

. Because

72 + 84 = 156

and

168 - 36 = 132

, the descriptive statements asserting that the sum can be

156

and the positive difference can be

132

are logically true.

Step-by-Step Solution

Use the property that the product of two numbers equals the product of their HCF and LCM.

A \times B = 12 \times 504 = 6048

This establishes the fundamental numerical relationship between the two integers.

Express

A

and

B

in terms of their HCF to simplify.

Let

A = 12x

and

B = 12y

, where

x

and

y

are coprime integers. Substituting into the product gives

144xy = 6048

, which simplifies to

xy = 42

Factoring out the HCF ensures that any pairs found will inherently respect the greatest common divisor requirement.

Find all coprime pairs

(x, y)

that multiply to 42 and calculate the corresponding integer pairs

\{A, B\}

The coprime pairs for

42

are

(1, 42)

(2, 21)

(3, 14)

, and

(6, 7)

. The corresponding

\{A, B\}

pairs are

\{12, 504\}

\{24, 252\}

\{36, 168\}

, and

\{72, 84\}

These pairs represent all mathematical possibilities before applying the specific contextual constraints.

Apply the problem's contextual condition that both integers must be strictly greater than 12.

The pair

\{12, 504\}

must be excluded. The valid pairs are exactly three:

\{24, 252\}

\{36, 168\}

, and

\{72, 84\}

Filtering out invalid mathematical solutions ensures strict adherence to the stated facts.

Evaluate the given statements against the three valid integer pairs.

The sum

72 + 84 = 156

validates the first statement. The difference

168 - 36 = 132

validates the second. The count of valid pairs is

3

, refuting the third statement. The LCM of

1/A

and

1/B

1/12

, refuting the fourth.

Systematically verifying which descriptive statements accurately reflect the discovered pairs.

Key Concept

Applying HCF and LCM properties alongside algebraic constraints to evaluate paired integers.

Estimated Time:2m 0s

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