Çözüldü Number of Positive Divisors

For any positive integer n, τ(n) represents the number of positive divisors of n. Which of the following is (are) true?

I. τ(5) = τ(7)
II. τ(5) · τ(7) = τ(35)
III. τ(5) + τ(7) = τ(12)
IV. τ(7) - τ(5) = 2

A) I only
B) I and II only
C) I and III only
D) I, II, and III
E) I and IV only

Correct Answer: B (I and II only)

Explanation:
Since 5 and 7 have two positive factors each, τ(5) = τ(7). (I is true.)
Since 35 has four divisors (1, 5, 7, and 35) and τ(5)·τ(7) = 2·2 = 4. (II is true.)
The value of τ(12) is 6, which is not equal to 2 + 2. (III is false.)
Since τ(5) = τ(7) is true, τ(7) - τ(5) = 0 ≠ 2. (IV is false.)

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