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modular:02_gcd [2014/01/31 11:45] marje |
modular:02_gcd [2014/02/01 00:00] marje |
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| - | {{ :modular:gcd.png?550 }} | + | {{ :modular:gcd.png?600 }} |
| We already know that common divisors of $20$ and $30$ are $10,\ 5,\ 2,\ 1$, thus the greatest common divisor is $10$. It is usually denoted $\gcd(20,30)=10$. | We already know that common divisors of $20$ and $30$ are $10,\ 5,\ 2,\ 1$, thus the greatest common divisor is $10$. It is usually denoted $\gcd(20,30)=10$. | ||
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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> | ||