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 $2520$ .Cevap
The Highest Common Factor (HCF) of $M$ and $N$ is $72$ .Cevap
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 $2520$ .

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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