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Zorluk: OrtaFactors, Multiples, and Prime Factorization

Consider the three fractions X=1621X = \frac{16}{21}, Y=815Y = \frac{8}{15}, and Z=2435Z = \frac{24}{35}. Based on the mathematical properties of factors and multiples for fractions, which of the following statements are correct? (Select all that apply)

  1. The Highest Common Factor (HCF) of the three fractions is 8105\frac{8}{105}.Cevap
  2. The Least Common Multiple (LCM) of the three fractions is 4848.Cevap
  3. C
    The Highest Common Factor (HCF) of the three fractions is 148\frac{1}{48}.
  4. D
    The Least Common Multiple (LCM) of the three fractions is 1058\frac{105}{8}.

Cevap

The correct statements are that the HCF of the fractions is 8/105 and the LCM of the fractions is 48.
The mathematical rules for fractions state that the HCF is determined by the ratio of the HCF of the numerators to the LCM of the denominators, which evaluates to 8/105. Similarly, the LCM is determined by the ratio of the LCM of the numerators to the HCF of the denominators, which evaluates to 48/1 or 48. The statements identifying these two specific values correctly represent these properties.

Adım Adım Çözüm

1
Verify that all given fractions are in their simplest form.
The fractions 16/21, 8/15, and 24/35 cannot be reduced further because their respective numerators and denominators have no common prime factors.
The standard formulas for finding the HCF and LCM of fractions strictly require all fractions to be reduced to their lowest terms first.
2
Extract the numerators and calculate their HCF and LCM.
Numerators are 16, 8, and 24. Their HCF is 8. Their LCM is 48.
These values are the necessary numerator components for the overarching fraction formulas.
3
Extract the denominators and calculate their HCF and LCM.
Denominators are 21, 15, and 35. Their HCF is 1 (since 3x7, 3x5, and 5x7 share no common prime factor across all three). Their LCM is 105 (3 x 5 x 7).
These values are the necessary denominator components for the overarching fraction formulas.
4
Apply the rule for the Highest Common Factor (HCF) of fractions.
HCF = HCF of Numerators / LCM of Denominators = 8 / 105.
This establishes the truth value for the statements regarding the HCF.
5
Apply the rule for the Least Common Multiple (LCM) of fractions.
LCM = LCM of Numerators / HCF of Denominators = 48 / 1 = 48.
This establishes the truth value for the statements regarding the LCM.

Anahtar Kavram

Calculating Highest Common Factor (HCF) and Least Common Multiple (LCM) for Fractions
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Konular

Factors, Multiples, and Prime Factorization

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