Euclidean Algorithm
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
先从这个心智模型开始
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
它如何工作，不堆砌不必要的术语
The key property is gcd(a, b) = gcd(b, a mod b).
It is much faster than checking every possible divisor.
把算法读成一连串决策
State the mathematical identity/property.
Choose numeric representation and constraints.
Translate the property into an iterative/recursive computation and analyze precision.
Read a tiny input by hand and write down the state after every important operation. For Euclidean Algorithm, the goal is to make the invariant visible rather than to memorize code.
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b !== 0) {
[a, b] = [b, a % b];
}
return a;
}不要死记语法。跟踪每一行中发生变化的状态，并把它与上面描述的不变量联系起来。
O(log min(a,b))
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
适用场景
- Use the direct mathematical property when it removes unnecessary brute force.
通常哪里会出错
- Ignoring floating-point precision or integer range.
- Using a formula outside its valid assumptions.
专业回答结构
Explain the invariant/property behind the modulo step; do not present it as a magic loop.
What problem does Euclidean Algorithm solve?
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
关于复杂度应该提到什么？
Its typical cost here is O(log min(a,b)), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
最常见的错误是什么？
Ignoring floating-point precision or integer range.