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modular:02_gcd [2014/01/31 11:41] marje |
modular:02_gcd [2014/01/31 19:15] marje |
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| The **least common multiple** of a set of numbers $\{a_1, a_2, \dots , a_n\}$ is denoted $$\operatorname{lcm}(a_1, a_2, \dots , a_n ).$$ | The **least common multiple** of a set of numbers $\{a_1, a_2, \dots , a_n\}$ is denoted $$\operatorname{lcm}(a_1, a_2, \dots , a_n ).$$ | ||
| It is a number $x$ such that | It is a number $x$ such that | ||
| - | * $a_i \mid x$ for all $i$, i.e., $x$ is a **common multiple** of $\{a_1, a_2,\dots , a_n\}$), | + | * $a_i \mid x$ for all $i$, i.e., $x$ is a **common multiple** of $\{a_1, a_2,\dots , a_n\}$, |
| * $x \mid y$ for any other common multiple $y$ of $\{a_1, a_2, \dots , a_n \}$. | * $x \mid y$ for any other common multiple $y$ of $\{a_1, a_2, \dots , a_n \}$. | ||
| </box> | </box> | ||
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| </WRAP> | </WRAP> | ||
| - | Clearly $\min(e_i, e'_i) +\max(e_i, e'_i)=e_i+e'_1$ (for example, $\min\{4,10\}+\max\{4,10\}=4+10=14$), thus obviously. | + | Clearly $\min(e_i, e'_i) +\max(e_i, e'_i)=e_i+e'_1$ (for example, $\min\{4,10\}+\max\{4,10\}=4+10=14$), thus |
| + | |||
| + | $$\gcd(a,b)\cdot\operatorname{lcm}(a,b)=a\cdot b.$$ | ||
| <WRAP nl> | <WRAP nl> | ||
| [[modular:04_euclidean_algorithm]]\\ | [[modular:04_euclidean_algorithm]]\\ | ||
| </WRAP> | </WRAP> | ||