Final answer:
The student's question is about the Euler's totient function ϕ(n). True statements regarding ϕ(n) and its properties in relation to multiplication with prime numbers are evaluated based on Proposition H. Statements 2, 4, and 5 are true based on the properties of the totient function.
Step-by-step explanation:
The student's question pertains to the Euler's totient function ϕ(n), which counts the positive integers up to a given integer n that are relatively prime to n. Proposition H likely refers to a property of the totient function regarding the multiplication of n by a prime not dividing n, which states ϕ(pn) = (p-1)ϕ(n) if p is a prime and p does not divide n.
Considering the above proposition:
- Statement 1 is false because 3 is a prime that divides 12, so ϕ(3⋅ 12) ≠ 3ϕ(12).
- Statement 2 is true because ϕ(3⋅12) = 2ϕ(12), assuming 3 does not divide 12.
- Statement 3 is false because 5 is a prime that does not divide 12, but ϕ(5⋅ 12) = 4ϕ(12) as per Proposition H, not 5ϕ(12).
- Statement 4 is true because 5 is a prime, does not divide 12, hence ϕ(5⋅ 12) = 4ϕ(12).
- Statement 5 is true because 25 is not prime, but ϕ(25⋅ 12) = 24ϕ(12) since 25 is 5 squared and 5 is a prime not dividing 12.
- Statement 6 is false since 35 is the product of 5 and 7, both primes that do not divide 12, so ϕ(35⋅ 12) would be the product of their respective ϕ values times ϕ(12), which is not simply 24ϕ(12).