🧩 因数与倍数：拼矩形与跳格子🧩 Factors & Multiples: Rectangles and Skip Counting
上一片我们拆开看：12 = 3 × 4。这一片追问：3、4 和 12 到底是什么关系？答案有两副面孔——因数与倍数，其实是同一件事的两面。Last leaf we cracked a number open: 12 = 3 × 4. This leaf asks what 3, 4 and 12 really mean to each other. The answer has two faces — factors and multiples — two sides of one coin.
第 1 步：因数是矩形的边Step 1: Factors are the sides of a rectangle
12 块方积木，能摆出几种整齐的矩形？How many neat rectangles can 12 square tiles make?
每排 3 块、摆 4 排，12 块刚好用完——3 和 4 都是 12 的因数。换成每排 2 块、摆 6 排也行；1 排 12 个也行；甚至 12 排每排 1 个也行。因数就是能把 12 恰好分完、不剩零头的数。Three tiles per row, four rows — all 12 used up. Both 3 and 4 are factors of 12. Two per row works too, and so does one long row of 12 — or twelve rows of one. Factors divide 12 exactly, with no leftovers.
同一个 12，换一个方向看，故事就反过来了。Look at the same 12 from the other side — the story flips.
第 2 步：倍数就是跳格子Step 2: Multiples are hops on the line
从 0 出发，每次跳 4 格：4、8、12——每一步都落在整格上，它们都是 4 的倍数。因为 12 = 3 × 4，12 既是 3 的倍数，也是 4 的倍数。因数把数拼起来，倍数从这个数往外跳。Start at 0 and hop 4 spaces at a time: 4, 8, 12 — every landing is a whole step. These are multiples of 4. Since 12 = 3 × 4, twelve is a multiple of both 3 and 4. Factors build a number; multiples bounce outward from it.
因数和倍数，生活里什么时候真的用得上？When do factors and multiples really show up in life?
第 3 步：排队分组、赶公交Step 3: Lining up and catching buses
12 个机器人排队：3 排每排 4 个，整齐；2 排每排 6 个，也整齐——这是用 12 的因数来分组。公交车每 6 分钟一班，12 分钟后第 2 班到——这是数 6 的倍数。分组找因数，赶车数倍数。Twelve robots line up: 3 rows of 4, neat; 2 rows of 6, also neat — that's grouping by factors of 12. A bus every 6 minutes means the second bus comes at minute 12 — that's counting multiples of 6. Grouping uses factors; schedules count multiples.
🎮 你来拼矩形（30 秒）🎮 Your turn: build the rectangles (30 seconds)
右边是 12 块积木。点出所有能把 12 拼成整齐矩形的数。Twelve tiles are waiting. Tap every number that tiles 12 into a neat rectangle.
一句话记住它：因数是“能把一个数分完的数”，倍数是“从这个数出发跳到的格子”。A = B × C 时，B 和 C 是 A 的因数，A 是 B 和 C 的倍数。Remember it in one line: factors divide a number exactly; multiples are the steps it lands on. If A = B × C, then B and C are factors of A, and A is a multiple of B and C.
因数能整除它：12 的因数有 1、2、3、4、6、12Factors divide exactly: 1, 2, 3, 4, 6, 12 are the factors of 12倍数是跳格子：4 的倍数有 4、8、12、16……Multiples are hops: 4, 8, 12, 16... are multiples of 4同一个式子两面看：12 = 3 × 4，3 和 4 是因数，12 是倍数One equation, two views: factors build it; the product is the multiple
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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