answer:The Euler totient function φ(n) is multiplicative because of the way it interacts with the prime factorization of n. Specifically, if n can be factored into two relatively prime numbers, say n = a*b, where gcd(a, b) = 1, then the totient function satisfies the following property: φ(n) = φ(a*b) = φ(a)*φ(b) To understand why this is true, consider the numbers that are relatively prime to n. These are precisely the numbers that are not divisible by any prime factor of n. Since a and b are relatively prime, their prime factorizations are disjoint, meaning they have no common prime factors. Let's count the numbers less than or equal to n that are relatively prime to n. We can do this by counting the numbers that are relatively prime to a, and then counting the numbers that are relatively prime to b, and then combining these two counts. The numbers that are relatively prime to a are precisely the numbers that are not divisible by any prime factor of a. Similarly, the numbers that are relatively prime to b are precisely the numbers that are not divisible by any prime factor of b. Since the prime factorizations of a and b are disjoint, the numbers that are relatively prime to a*b (i.e., to n) are precisely the numbers that are relatively prime to both a and b. Therefore, we can count these numbers by multiplying the number of numbers relatively prime to a and the number of numbers relatively prime to b. Mathematically, this can be represented as: φ(a*b) = |{1 ≤ x ≤ a*b : gcd(x, a*b) = 1}| = |{1 ≤ x ≤ a : gcd(x, a) = 1}| * |{1 ≤ x ≤ b : gcd(x, b) = 1}| = φ(a) * φ(b) This property holds for any positive integers a and b that are relatively prime, which is why the Euler totient function is said to be multiplicative.

answer:(1) Since a_3 is the geometric mean of a_1 and a_9, and {a_n} is an arithmetic sequence, it holds that a_3^2 = a_1 cdot a_9. With a_1 = 2, this translates into the equation (a_1 + 2d)^2 = a_1(a_1 + 8d). Substituting a_1 = 2, we get (2 + 2d)^2 = 2 cdot (2 + 8d), which simplifies to 4d^2 + 8d + 4 = 16 + 16d. Further simplification lead to d^2 - 2d = 0. Since we are given that d neq 0, dividing through by d we find d - 2 = 0, hence d = 2. The general term of an arithmetic sequence is given by a_n = a_1 + (n-1)d. Substituting a_1 = 2 and d = 2, we get the general term a_n = 2 + (n-1) cdot 2 = 2n. So the answer for the first part is boxed{a_n = 2n}. (2) Knowing the general term a_n = 2n, we rewrite the sequence as frac {1}{a_{n}a_{n+1}} = frac {1}{2n cdot (2n+2)} = frac {1}{4n(n+1)}. This can be further expressed as a telescoping series: frac {1}{4n(n+1)} = frac {1}{4}left(frac {1}{n} - frac {1}{n+1}right). Therefore, the sum of the first n terms T_n becomes: T_n = frac {1}{4}left(1 - frac {1}{2} + frac {1}{2} - frac {1}{3} + ldots + frac {1}{n} - frac {1}{n+1}right). All terms cancel out except for the first and the last, yielding T_n = frac {1}{4}left(1 - frac {1}{n+1}right) = frac {1}{4}frac {n}{n+1}. So, for the second part, the sum is boxed{T_n = frac {n}{4(n+1)}}.

answer:Kyle purchased 2 glass bottles, each capable of holding 15 origami stars, so the total capacity of the first 2 bottles is 2 * 15 = 30 origami stars. He later bought 3 more identical bottles, so the total number of bottles he purchased is 2 + 3 = 5 bottles. If each bottle has a capacity of 15 origami stars, then the total capacity of the 5 bottles is 5 * 15 = 75 origami stars. Therefore, Kyle needs to make 75 origami stars in order to fill all the glass bottles he purchased. 75 The answer is: 75

answer:Starting a thriller novel can be as thrilling as the story itself. Consider beginning with a hook that grabs the reader's attention and sets the tone for the rest of the book. This could be a dramatic event, an ominous setting, or an intriguing character introduction. Imagine opening your story on a stormy night, with your protagonist driving down a deserted road, their GPS leading them further and further away from civilization. Suddenly, their car sputters and dies, plunging them into darkness. As they step out into the torrential rain, they stumble upon a mysterious figure or a cryptic message that sets the wheels of the plot in motion. Alternatively, you could introduce your protagonist in a more mundane setting, but with a sense of foreboding or unease. Perhaps they're a detective who's just received a disturbing phone call or a cryptic letter that hints at a long-buried secret. As they begin to investigate, they're drawn into a web of intrigue that threatens to upend their entire world. The key is to create tension and suspense from the very beginning, while also introducing the reader to your protagonist and setting. Avoid info dumping or lengthy exposition – instead, focus on crafting a scene that's both atmospheric and propulsive. One technique to consider is starting your story in medias res, or in the middle of the action. This can be an effective way to drop the reader into the midst of a thrilling scene, while also raising questions that will keep them turning the pages. For example, you might begin with your protagonist fleeing from danger, without revealing what they're running from or why. As the story unfolds, the reader will be piecing together the clues alongside the protagonist, which can create a sense of urgency and investment.