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1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

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Mathematics · 术语表

什么是 Least common multiple？

Definition 6.10 University Mathematics — Year 1 · 第 6 章 — Integer Arithmetic

$\operatorname{lcm}(a, b)$ is the generator in $\N$ of the subgroup $a\Z \cap b\Z$ : it is a common multiple of $a$ and $b$ that divides every common multiple, and for $a, b \in \N^*$ ,

\gcd(a,b) \times \operatorname{lcm}(a,b) = ab \qquad (\text{proof in } \text{Exercise 6.5}).

示例

Example 6.11 (Alignment problems are lcm problems)

Two meshed gears have $84$ and $36$ teeth. After how many teeth of common motion do they return to their initial position together? The configuration repeats when the number of elapsed teeth is a common multiple of $84$ and $36$ ; the first time is

\operatorname{lcm}(84, 36) = \frac{84 \times 36}{\gcd(84, 36)} = \frac{3024}{12} = 252

teeth — that is, $3$ turns of the large gear and $7$ of the small one ( $252/84$ and $252/36$ ). Note the practical route: compute the gcd first (Euclid: $84 = 2\times36 + 12$ , $36 = 3\times12$ ), then divide — never build the lcm by listing multiples. Every periodic-coincidence question (gears, planetary alignments, repeating decimals meeting) reduces to this one computation.

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