Consider the positive integers M = 360 and N = 504. Based on the mathematical properties o… — State PSC Exam

Consider the positive integers $M = 360$ and $N = 504$ . Based on the mathematical properties of factors and multiples, which of the following statements are correct?

The Least Common Multiple (LCM) of $M$ and $N$ is $2520$ .Answer
The Highest Common Factor (HCF) of $M$ and $N$ is $72$ .Answer
C
The integer $M$ has exactly $24$ distinct positive prime factors.
D
The Highest Common Factor (HCF) of the fractions $\frac{M}{7}$ and $\frac{N}{5}$ is $2520$ .

Answer

The statements confirming the Least Common Multiple (LCM) is 2520 and the Highest Common Factor (HCF) is 72 are mathematically correct.

Based on the prime factorizations

M = 2^3 \times 3^2 \times 5

and

N = 2^3 \times 3^2 \times 7

, the HCF is derived from the lowest common powers (

2^3 \times 3^2 = 72

), and the LCM is derived from the highest powers of all prime bases (

72 \times 5 \times 7 = 2520

). Thus, the statements asserting these two values are correct.

Step-by-Step Solution

Determine the prime factorizations of M and N.

M = 2^3 \times 3^2 \times 5^1

and

N = 2^3 \times 3^2 \times 7^1

Prime factorization is the foundational step required to find factors, HCF, and LCM accurately.

Calculate the Highest Common Factor (HCF) and Least Common Multiple (LCM) of M and N.

HCF =

2^3 \times 3^2 = 72

. LCM =

2^3 \times 3^2 \times 5 \times 7 = 2520

This confirms the validity of the first two statements.

Evaluate the number of factors for integer M.

Total positive factors =

(3+1)(2+1)(1+1) = 24

. However, distinct prime factors are only 2, 3, and 5 (total of 3).

This shows the statement claiming 24 prime factors is factually incorrect due to a specific modifier (prime).

Calculate the HCF of the given fractions.

HCF =

\frac{\text{HCF}(360, 504)}{\text{LCM}(7, 5)} = \frac{72}{35}

This invalidates the final statement, which improperly applied the LCM fraction formula.

Key Concept

Applying prime factorization to calculate HCF, LCM, total factor counts, and properties of fractions.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic