Set Fn gives all factors of n Set Mn gives all multiples of n less than 1000 Which of the following ...........

Quantitative Aptitude ( CCS) Set theory

Set F_n gives all factors of n. Set M_n gives all multiples of n less than 1000. Which of the following statements is/are true?

i. F₁₀₈ ∩ F₈₄ = F₁₂

ii. M₁₂ ∪ M₁₈ = M₃₆

iii. M₁₂ ∩ M₁₈ = M₃₆

iv. M₁₂ ⊂ (M₆ ∩ M₄)

i, ii and iii only

i, iii and iv only

i and iii only

All statements are true

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Explanation:

i. F₁₀₈ ∩ F₈₄
This is the set of all numbers that are factors of both 108 and 84
=> this is set of all common factors of 84 and 108
=> this is set of numbers that are factors of the Highest Common Factor of 84 and 108.
HCF (84, 108) = 12.
F₁₀₈ ∩ F₈₄ = F₁₂ - this is true.
ii. M₁₂ will have numbers {12, 24, 36, 48, ….} Numbers like 12, 24, … will not feature in M₃₆.
So, Statement B cannot be true.
iii. M₁₂ ∩ M₁₈ - this is the set of all common multiples of 12 and 18.
This is the set of numbers that are multiples of the Least Common Multiple of 12 and 18. (which is 36).
This statement is also true.
iv. Using the same logic as that used in statement iii, we can determine that statement iv is also true. Remember that every set is a subset of itself.
Hence, the answer is i, iii and iv are true.