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Omni-MATH / Let n be the product of the first 10 primes, and let S=∑_(xy∣n)φ(x)⋅y where φ(x) denotes the number of positive integers less than or equal to x that are relatively prime to x, and…

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problem

Let nn be the product of the first 10 primes, and let S=∑xy∣nφ(x)⋅yS=\sum_{x y \mid n} \varphi(x) \cdot y where φ(x)\varphi(x) denotes the number of positive integers less than or equal to xx that are relatively prime to xx, and the sum is taken over ordered pairs (x,y)(x, y) of positive integers for which xyx y divides nn. Compute Sn\frac{S}{n}.
Plain-text mathematical notation (without MathML)
Let n be the product of the first 10 primes, and let S=∑_(xy∣n)φ(x)⋅y where φ(x) denotes the number of positive integers less than or equal to x that are relatively prime to x, and the sum is taken over ordered pairs (x,y) of positive integers for which xy divides n. Compute (S)/(n).
Original LaTeX notation
Let $n$ be the product of the first 10 primes, and let $$S=\sum_{x y \mid n} \varphi(x) \cdot y$$ where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $x y$ divides $n$. Compute $\frac{S}{n}$.

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Source and history

task_id
1e2c0855-2df7-5adc-acd8-ab4406aff151
task_key
test--1e2c0855-2df7-5adc-acd8-ab4406aff151
task_revision_id
1
config / split
/ test

Official source

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