The sum of all positive factors of a number is 124. Which statement is correct?
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The sum of all positive factors of a number is 124. Which statement is correct?
- A.
Exactly one such number exists, and it is between 40 and 50
- B.
Exactly one such number exists, and it is between 50 and 60
- C.
Exactly one such number exists, and it is between 60 and 80
- D.
More than one such number exists
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Correct answer: D
Correct answer: More than one such number exists
A useful method is to use prime factorization and the divisor-sum formula. If n = pᵃqᵇ..., then the sum of all positive factors is σ(n) = (1 + p + ... + pᵃ)(1 + q + ... + qᵇ)... .
For 48: 48 = 2⁴ × 3, so σ(48) = (1 + 2 + 4 + 8 + 16)(1 + 3) = 31 × 4 = 124.
For 75: 75 = 3 × 5², so σ(75) = (1 + 3)(1 + 5 + 25) = 4 × 31 = 124.
Conclusion: both 48 and 75 satisfy the condition, so the number is not unique.
Final answer: More than one such number exists.