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<title>GCD & LCM - 核心性质与公式</title>
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</head>
<body>
<div id="app">
<div class="content-area">
<!-- Page 1: 四大性质 -->
<div v-show="currentPage === 1">
<div class="top-tabs">
<div class="top-tab" :class="{active: propertyTab === 'product'}" @click="propertyTab = 'product'">
乘积关系
</div>
<div class="top-tab" :class="{active: propertyTab === 'coprime'}" @click="propertyTab = 'coprime'">
互质关系
</div>
<div class="top-tab" :class="{active: propertyTab === 'multiple'}" @click="propertyTab = 'multiple'">
倍数关系
</div>
<div class="top-tab" :class="{active: propertyTab === 'scale'}" @click="propertyTab = 'scale'">
同倍缩放
</div>
</div>
<div class="page-container">
<!-- 乘积关系 -->
<div v-show="propertyTab === 'product'">
<div class="section-title">🎯 性质1：乘积关系</div>
<div class="formula-box">
<div class="formula-title">★ 黄金公式 ★</div>
<div class="formula-content">
$A \times B = \text{GCD}(A, B) \times \text{LCM}(A, B)$
</div>
</div>
<div class="card">
<div class="tag tag-blue">公式含义</div>
<p style="font-size: 14px; color: #334155; line-height: 1.8; margin-top: 10px;">
两个数的<strong>乘积</strong> = 它们的<strong>最大公约数</strong> × <strong>最小公倍数</strong>
</p>
<div style="background: #FEF3C7; padding: 12px; border-radius: 8px; margin-top: 10px;">
<strong style="color: #92400E;">💡 记忆技巧：</strong><br>
知道三个，就能求第四个！
</div>
</div>
<div class="card">
<div class="tag tag-purple">动画演示</div>
<div class="step-controls">
<button class="step-btn" @click="animateProduct">▶️ 演示验证</button>
</div>
<div class="anim-stage">
<div class="visual-demo" id="product-demo">
<!-- 动画将在这里生成 -->
</div>
<div class="anim-text" id="product-text">点击按钮开始演示</div>
</div>
</div>
<div class="card">
<div class="tag tag-green">应用场景</div>
<div style="font-size: 13px; color: #334155; line-height: 1.8;">
<p><strong>📌 典型题型：</strong></p>
<div style="background: #EFF6FF; padding: 12px; border-radius: 6px; margin: 10px 0;">
• 已知两数乘积和GCD，求LCM<br>
• 已知两数乘积和LCM，求GCD<br>
• 已知一数、GCD和LCM，求另一数
</div>
</div>
</div>
<button class="speak-btn" @click="speakProduct">
🔊 听讲解
</button>
</div>
<!-- 互质关系 -->
<div v-show="propertyTab === 'coprime'">
<div class="section-title">🤝 性质2：互质关系</div>
<div class="formula-box">
<div class="formula-title">★ 互质公式 ★</div>
<div class="formula-content">
若 $\text{GCD}(A, B) = 1$，则 $\text{LCM}(A, B) = A \times B$
</div>
</div>
<div class="card">
<div class="tag tag-blue">什么是互质？</div>
<p style="font-size: 14px; color: #334155; line-height: 1.8; margin-top: 10px;">
两个数除了1以外，没有别的<strong>公约数</strong>，我们就说这两个数<strong>互质</strong>。
</p>
<div style="background: #F0FDF4; padding: 12px; border-radius: 8px; margin-top: 10px;">
<strong style="color: #065F46;">✅ 互质的例子：</strong><br>
• 3 和 5（相邻质数）<br>
• 8 和 9（相邻自然数）<br>
• 7 和 15（一个质数和一个合数）
</div>
</div>
<div class="card">
<div class="tag tag-purple">动画演示</div>
<div class="step-controls">
<button class="step-btn" @click="animateCoprime">▶️ 演示互质性质</button>
</div>
<div class="anim-stage">
<div class="visual-demo" id="coprime-demo">
<!-- 动画将在这里生成 -->
</div>
<div class="anim-text" id="coprime-text">点击按钮开始演示</div>
</div>
</div>
<div class="card">
<div class="tag tag-orange">特殊情况</div>
<div style="font-size: 13px; color: #334155; line-height: 1.8;">
<p><strong>🎯 一定互质的情况：</strong></p>
<div style="background: #FEF3C7; padding: 12px; border-radius: 6px; margin: 10px 0;">
1️⃣ <strong>相邻的两个自然数</strong>（如7和8）<br>
2️⃣ <strong>两个不同的质数</strong>（如3和7）<br>
3️⃣ <strong>1和任何数</strong>（1的特殊性）
</div>
</div>
</div>
<button class="speak-btn" @click="speakCoprime">
🔊 听讲解
</button>
</div>
<!-- 倍数关系 -->
<div v-show="propertyTab === 'multiple'">
<div class="section-title">📊 性质3：倍数关系</div>
<div class="formula-box">
<div class="formula-title">★ 倍数公式 ★</div>
<div class="formula-content">
若 $A$ 是 $B$ 的倍数，则<br>
$\text{GCD}(A, B) = B$（较小的）<br>
$\text{LCM}(A, B) = A$（较大的）
</div>
</div>
<div class="card">
<div class="tag tag-blue">公式含义</div>
<p style="font-size: 14px; color: #334155; line-height: 1.8; margin-top: 10px;">
如果 $A = B \times k$（$A$是$B$的$k$倍），那么：
</p>
<div style="background: #EFF6FF; padding: 12px; border-radius: 8px; margin: 10px 0;">
• <strong>最大公约数</strong>就是小的那个数$B$<br>
• <strong>最小公倍数</strong>就是大的那个数$A$
</div>
</div>
<div class="card">
<div class="tag tag-purple">动画演示</div>
<div class="step-controls">
<button class="step-btn" @click="animateMultiple">▶️ 演示倍数关系</button>
</div>
<div class="anim-stage">
<div class="visual-demo" id="multiple-demo">
<!-- 动画将在这里生成 -->
</div>
<div class="anim-text" id="multiple-text">点击按钮开始演示</div>
</div>
</div>
<div class="card">
<div class="tag tag-green">理解要点</div>
<div style="font-size: 13px; color: #334155; line-height: 1.8;">
<p><strong>🎯 为什么是这样？</strong></p>
<div style="background: #F0FDF4; padding: 12px; border-radius: 6px; margin: 10px 0;">
因为大数$A$已经"包含"了小数$B$，所以：<br>
• 它们共同拥有的最大零件就是$B$<br>
• 能同时装下它们的最小仓库就是$A$
</div>
</div>
</div>
<button class="speak-btn" @click="speakMultiple">
🔊 听讲解
</button>
</div>
<!-- 同倍缩放 -->
<div v-show="propertyTab === 'scale'">
<div class="section-title">🔄 性质4：同倍缩放</div>
<div class="formula-box">
<div class="formula-title">★ 缩放公式 ★</div>
<div class="formula-content">
$\text{GCD}(kA, kB) = k \times \text{GCD}(A, B)$<br>
$\text{LCM}(kA, kB) = k \times \text{LCM}(A, B)$
</div>
</div>
<div class="card">
<div class="tag tag-blue">公式含义</div>
<p style="font-size: 14px; color: #334155; line-height: 1.8; margin-top: 10px;">
两个数同时<strong>乘以</strong>或<strong>除以</strong>同一个数$k$，它们的GCD和LCM也会同步变化$k$倍。
</p>
<div style="background: #FEF3C7; padding: 12px; border-radius: 8px; margin-top: 10px;">
<strong style="color: #92400E;">💡 神奇用途：</strong><br>
算大数时先缩小，算完再放大回去！
</div>
</div>
<div class="card">
<div class="tag tag-purple">动画演示</div>
<div class="step-controls">
<button class="step-btn" @click="animateScale">▶️ 演示缩放过程</button>
</div>
<div class="anim-stage">
<div class="calc-demo" id="scale-demo">
<!-- 动画将在这里生成 -->
</div>
<div class="anim-text" id="scale-text">点击按钮开始演示</div>
</div>
</div>
<div class="card">
<div class="tag tag-green">实战技巧</div>
<div style="font-size: 13px; color: #334155; line-height: 1.8;">
<p><strong>📌 例如求GCD(1200, 1600)：</strong></p>
<div class="step-by-step">
<strong>步骤1：</strong>观察发现两数都是100的倍数<br>
<strong>步骤2：</strong>同时除以100 → 变成GCD(12, 16)<br>
<strong>步骤3：</strong>计算GCD(12, 16) = 4<br>
<strong>步骤4：</strong>乘回100 → GCD(1200, 1600) = 400
</div>
</div>
</div>
<button class="speak-btn" @click="speakScale">
🔊 听讲解
</button>
</div>
</div>
</div>
<!-- Page 2: 5道经典例题 -->
<div v-show="currentPage === 2">
<div class="top-tabs">
<div class="top-tab" :class="{active: exampleIndex === index}"
v-for="(ex, index) in classicExamples" :key="'tab-' + index"
@click="switchExample(index)">
例题{{ index + 1 }}
</div>
</div>
<div class="page-container">
<div v-for="(ex, index) in classicExamples" :key="'example-' + index"
v-show="exampleIndex === index">
<div class="section-title">{{ ex.title }}</div>
<div class="card">
<div class="tag tag-blue">题目</div>
<div class="question-text" v-html="ex.question"></div>
</div>
<div class="card">
<div class="tag tag-purple">分步解析</div>
<div class="step-controls">
<button class="step-btn" @click="prevExampleStep(index)" :disabled="exampleSteps[index] === 0">
◀ 上一步
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<button class="step-btn" @click="nextExampleStep(index)" :disabled="exampleSteps[index] >= ex.steps.length - 1">
下一步 ▶
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<button class="step-btn" @click="resetExampleStep(index)" style="background: #64748B;">
🔄 重置
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</div>
<div class="anim-stage">
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:style="{opacity: exampleSteps[index] >= sIndex ? 1 : 0}">
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<div class="calc-explanation">{{ step.explanation }}</div>
</div>
</div>
<div class="anim-text">
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</div>
<div class="card">
<div class="tag tag-green">答案</div>
<div class="example-solution" v-html="ex.answer"></div>
</div>
</div>
</div>
</div>
<!-- Page 3: 10道练习题 -->
<div v-show="currentPage === 3" class="page-container">
<div class="section-title">✏️ 10道练习题</div>
<div v-if="!practiceDone">
<div class="question-card">
<div class="tag tag-blue">练习 {{ currentPracticeIndex + 1 }}/10</div>
<div class="question-text" v-html="practiceQuestions[currentPracticeIndex].question"></div>
<input
type="text"
class="answer-input"
v-model="userAnswer"
placeholder="输入答案（可以是数字或分数，如7/11）"
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<strong>💡 提示：</strong>这些题目来自华杯赛、希望杯、迎春杯等真题。<br>
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// 5道经典例题
classicExamples: [
{
title: '例题1：乘积公式的直接应用',
question: '已知两个自然数 $A$ 和 $B$，它们的乘积 $A \\times B = 432$，最大公约数 $\\text{GCD}(A, B) = 6$。<br>求这两个数的最小公倍数 $\\text{LCM}(A, B)$。',
steps: [
{
equation: '已知：$A \\times B = 432$',
explanation: '两数的乘积',
text: '第1步：明确已知条件'
},
{
equation: '已知：$\\text{GCD}(A, B) = 6$',
explanation: '两数的最大公约数',
text: '第2步：记录GCD的值'
},
{
equation: '使用公式：$A \\times B = \\text{GCD} \\times \\text{LCM}$',
explanation: '乘积关系公式',
text: '第3步：套用黄金公式'
},
{
equation: '$432 = 6 \\times \\text{LCM}$',
explanation: '代入数值',
text: '第4步：代入已知数值'
},
{
equation: '$\\text{LCM} = 432 \\div 6 =$ <span class="calc-highlight">72</span>',
explanation: '计算结果',
text: '第5步：计算得出答案'
}
],
answer: '<strong style="color: #8B5CF6; font-size: 18px;">答案：72</strong><br>通过乘积公式，只需简单的除法就能求出LCM！'
},
{
title: '例题2：互质关系的反向构造',
question: '两个自然数互质，它们的最小公倍数是 $35$。<br>求这两个数的和是多少？',
steps: [
{
equation: '已知：$\\text{GCD}(a, b) = 1$（互质）',
explanation: '两数互质',
text: '第1步：理解互质条件'
},
{
equation: '已知：$\\text{LCM}(a, b) = 35$',
explanation: '最小公倍数',
text: '第2步：记录LCM'
},
{
equation: '互质时：$\\text{LCM} = a \\times b$',
explanation: '互质性质',
text: '第3步：应用互质公式'
},
{
equation: '$a \\times b = 35$',
explanation: '转化为乘积问题',
text: '第4步：问题转化'
},
{
equation: '分解：$35 = 1 \\times 35 = 5 \\times 7$',
explanation: '找所有因数对',
text: '第5步：分解35'
},
{
equation: '检验互质：$(1, 35)$ 互质 ✓，$(5, 7)$ 互质 ✓',
explanation: '确认都满足互质条件',
text: '第6步：验证互质'
},
{
equation: '<span class="calc-highlight">和为：$1 + 35 = 36$</span> 或 <span class="calc-highlight">$5 + 7 = 12$</span>',
explanation: '计算两组和',
text: '第7步：计算各组和'
}
],
answer: '<strong style="color: #8B5CF6; font-size: 18px;">答案：36 或 12</strong><br>两种情况都符合题意：(1, 35) 或 (5, 7)'
},
{
title: '例题3：同倍缩放法算大数',
question: '求 $1200$ 和 $1600$ 的最大公约数和最小公倍数。',
steps: [
{
equation: '观察：$1200 = 12 \\times 100$，$1600 = 16 \\times 100$',
explanation: '提取公因数100',
text: '第1步：观察发现公因数'
},
{
equation: '缩小：除以100 → 求 $\\text{GCD}(12, 16)$',
explanation: '先算小数',
text: '第2步：同时缩小100倍'
},
{
equation: '$\\text{GCD}(12, 16) = 4$',
explanation: '计算小数的GCD',
text: '第3步：计算GCD(12,16)'
},
{
equation: '$\\text{LCM}(12, 16) = 48$',
explanation: '计算小数的LCM',
text: '第4步：计算LCM(12,16)'
},
{
equation: '还原：<span class="calc-highlight">$\\text{GCD}(1200, 1600) = 4 \\times 100 = 400$</span>',
explanation: '乘回100',
text: '第5步：GCD还原放大'
},
{
equation: '还原：<span class="calc-highlight">$\\text{LCM}(1200, 1600) = 48 \\times 100 = 4800$</span>',
explanation: '乘回100',
text: '第6步：LCM还原放大'
}
],
answer: '<strong style="color: #8B5CF6; font-size: 18px;">答案：GCD=400，LCM=4800</strong><br>同倍缩放让大数计算变简单！'
},
{
title: '例题4：比例与公约数',
question: '两个数的比是 $3:4$，它们的最大公约数是 $5$。<br>求这两个数的最小公倍数。',
steps: [
{
equation: '比例：$a : b = 3 : 4$',
explanation: '两数的比',
text: '第1步：理解比例关系'
},
{
equation: '设：$a = 3k$，$b = 4k$',
explanation: 'k是公因数',
text: '第2步：用k表示两数'
},
{
equation: '已知：$\\text{GCD}(3k, 4k) = 5$',
explanation: '最大公约数条件',
text: '第3步：应用GCD条件'
},
{
equation: '由同倍性质：$\\text{GCD}(3k, 4k) = k \\times \\text{GCD}(3, 4)$',
explanation: '使用同倍缩放',
text: '第4步：套用同倍公式'
},
{
equation: '$k \\times 1 = 5$（因为3和4互质）',
explanation: 'GCD(3,4)=1',
text: '第5步：计算k值'
},
{
equation: '$k = 5$，所以 $a = 15$，$b = 20$',
explanation: '求出具体数值',
text: '第6步：确定两个数'
},
{
equation: '<span class="calc-highlight">$\\text{LCM}(15, 20) = 60$</span>',
explanation: '计算最小公倍数',
text: '第7步：计算LCM'
}
],
answer: '<strong style="color: #8B5CF6; font-size: 18px;">答案：60</strong><br>也可以用公式：$\\text{LCM} = 5 \\times \\text{LCM}(3, 4) = 5 \\times 12 = 60$'
},
{
title: '例题5：倍数关系的陷阱',
question: '$A$ 是 $B$ 的因数（且 $A \\neq B$），已知 $\\text{LCM}(A, B) = 30$。<br>求 $B$ 是多少？',
steps: [
{
equation: '已知：$A$ 是 $B$ 的因数',
explanation: '即B是A的倍数',
text: '第1步：理解倍数关系'
},
{
equation: '倍数关系性质：$\\text{LCM}(A, B) = B$（较大数）',
explanation: '关键公式',
text: '第2步：应用倍数性质'
},
{
equation: '已知：$\\text{LCM}(A, B) = 30$',
explanation: 'LCM的值',
text: '第3步：代入已知条件'
},
{
equation: '<span class="calc-highlight">直接得出：$B = 30$</span>',
explanation: '答案一步到位',
text: '第4步：得出答案'
}
],
answer: '<strong style="color: #8B5CF6; font-size: 18px;">答案：30</strong><br>倍数关系下，LCM就是较大的那个数！'
}
],
// 练习题
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practiceQuestions: [
{question: '已知 $a \\times b = 100$，$\\text{GCD}(a, b) = 5$，求 $\\text{LCM}(a, b)$', answer: '20', explanation: '使用公式：$\\text{LCM} = (a \\times b) \\div \\text{GCD} = 100 \\div 5 = 20$'},
{question: '已知 $\\text{GCD}(a, b) = 1$（互质），且 $a \\times b = 20$，求 $\\text{LCM}(a, b)$', answer: '20', explanation: '互质时 $\\text{LCM} = a \\times b = 20$'},
{question: '求 $\\text{LCM}(300, 500)$（提示：先缩放）', answer: '1500', explanation: '提取100：$100 \\times \\text{LCM}(3, 5) = 100 \\times 15 = 1500$'},
{question: '已知 $A$ 是 $B$ 的倍数，且 $A=24$，求 $\\text{LCM}(A, B)$', answer: '24', explanation: '倍数关系，$\\text{LCM}$ 是较大的数 $A = 24$'},
{question: '已知 $\\text{GCD}(a, b) = 10$，$\\text{LCM}(a, b) = 100$，求 $a \\times b$', answer: '1000', explanation: '乘积 $= \\text{GCD} \\times \\text{LCM} = 10 \\times 100 = 1000$'},
{question: '判断：两个数的最小公倍数一定是它们最大公约数的倍数（输入"对"或"错"）', answer: '对', explanation: '正确！$\\text{LCM}$ 包含所有质因数，$\\text{GCD}$ 是公共质因数，必然包含'},
{question: '若 $a = 3b$，则 $\\text{GCD}(a, b)$ 等于多少？（用b表示）', answer: 'b', explanation: '因为 $a$ 是 $b$ 的倍数，$\\text{GCD}$ 是较小数 $b$'},
{question: '已知两个质数 $P$ 和 $Q$，求 $\\text{LCM}(P, Q)$（用P和Q表示，如P*Q）', answer: 'P*Q', explanation: '质数互质，$\\text{LCM} = P \\times Q$'},
{question: '求 $\\text{GCD}(150, 250)$（提示：同倍缩放）', answer: '50', explanation: '$50 \\times \\text{GCD}(3, 5) = 50 \\times 1 = 50$'},
{question: '已知两数乘积是60，最小公倍数是30，求最大公约数', answer: '2', explanation: '$\\text{GCD} = (a \\times b) \\div \\text{LCM} = 60 \\div 30 = 2$'}
],
// 杯赛真题
currentOlympiadIndex: 0,
olympiadScore: 0,
olympiadAnswered: false,
isOlympiadCorrect: false,
userOlympiadAnswer: '',
olympiadDone: false,
olympiadQuestions: [
{
question: '（希望杯 真题）已知两个自然数 $A$ 和 $B$ 的乘积是 $192$，它们的最大公约数是 $4$。求它们的最小公倍数。',
answer: '48',
hint: '使用乘积公式',
explanation: '<strong>解析：</strong><strong>$\\text{LCM} = (A \\times B) \\div \\text{GCD} = 192 \\div 4 = 48$</strong>'
},
{
question: '（华杯赛 真题）两个自然数的最大公约数是 $6$，最小公倍数是 $72$。已知其中一个数是 $18$，另一个数是多少？',
answer: '24',
hint: '使用公式 A×B = GCD×LCM',
explanation: '<strong>解析：</strong><strong>$18 \\times B = 6 \\times 72$ → $B = 432 \\div 18 = 24$</strong>'
},
{
question: '（迎春杯 真题）已知 $a$ 和 $b$ 互质，且 $\\text{LCM}(a, b) = 63$。求 $a$ 和 $b$ 的值可能有几组？（不考虑顺序）',
answer: '2',
hint: '互质时LCM = a×b，分解63',
explanation: '<strong>解析：</strong>$a \\times b = 63 = 3^2 \\times 7$<br>互质组合：$(1, 63)$ 和 $(7, 9)$<br>共<strong>2组</strong>'
},
{
question: '（走美杯 真题）两个数的最大公约数是 $15$，最小公倍数是 $180$。这两个数相差 $15$，求这两个数。',
answer: '45,60',
hint: '设两数为15x和15y，其中x,y互质',
explanation: '<strong>解析：</strong>设 $a=15x, b=15y$（$x,y$ 互质）<br>$15xy = 180$ → $xy = 12$<br>$15(x-y) = 15$ → $x-y = 1$<br>积12差1的互质数：$(3, 4)$<br>数是 <strong>15 \\times 3 = 45$ </strong>和 $15 \\times 4 = <strong>60</strong>$'
},
{
question: '（中环杯 真题）已知 $a, b$ 都是两位数，且 $\\text{GCD}(a, b) = 6$，$\\text{LCM}(a, b) = 90$。求 $a + b$ 的值。',
answer: '48',
hint: '用乘积公式求a×b，再分解',
explanation: '<strong>解析：</strong>$a \\times b = 6 \\times 90 = 540$<br>设 $a=6x, b=6y$（$x,y$ 互质）<br>$36xy = 540$ → $xy = 15$<br>互质对：$(3, 5)$<br>数是 $18$ 和 $30$<br>和 $= 18 + 30 = <strong>48</strong>$'
},
{
question: '（高难度）已知 $\\text{GCD}(x, y) = 5$，$\\text{LCM}(x, y) = 120$。求 $x$ 和 $y$ 的和最小是多少？',
answer: '55',
hint: '求xy，再分解成互质对',
explanation: '<strong>解析：</strong>$xy = 5 \\times 120 = 600$<br>设 $x=5m, y=5n$（$m,n$ 互质）<br>$25mn = 600$ → $mn = 24$<br>互质对：$(1,24)$和为125，$(3,8)$和为55<br>最小和 $= <strong>55</strong>$'
},
{
question: '（思维题）$A$ 和 $B$ 的最大公约数是 $12$，$A$ 和 $B$ 的和是 $60$。求 $\\text{LCM}(A, B)$。',
answer: '48或72',
hint: '设A=12m, B=12n',
explanation: '<strong>解析：</strong>设 $A=12m, B=12n$<br>$12(m+n) = 60$ → $m+n = 5$<br>互质对：$(1,4)$或$(2,3)$<br>$(1,4)$：$A=12, B=48$，$\\text{LCM}=48$<br>$(2,3)$：$A=24, B=36$，$\\text{LCM}=72$'
},
{
question: '（奥数经典）两数乘积 $3150$，最大公约数 $15$。这两数之差最小是多少？',
answer: '75',
hint: '设两数为15x和15y',
explanation: '<strong>解析：</strong>设 $a=15x, b=15y$<br>$225xy = 3150$ → $xy = 14$<br>互质对：$(1,14)$差195，$(2,7)$差75<br>最小差 $= 15 \\times 5 = <strong>75</strong>$'
},
{
question: '（挑战题）已知 $a, b, c$ 互不相等，$\\text{GCD}(a,b)=\\text{GCD}(b,c)=\\text{GCD}(a,c)=15$。若 $a \\times b \\times c = 33750$，求 $\\text{LCM}(a,b,c)$。',
answer: '150',
hint: '设a=15x, b=15y, c=15z',
explanation: '<strong>解析：</strong>设 $a=15x, b=15y, c=15z$<br>$15^3 \\cdot xyz = 33750$ → $xyz = 10 = 1 \\times 2 \\times 5$<br>$a=15, b=30, c=75$<br>$\\text{LCM} = 15 \\times 10 = <strong>150</strong>$'
}
]
}
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speakProduct() {
this.speak('乘积关系是最重要的公式：两个数的乘积等于它们的最大公约数乘以最小公倍数。知道三个，就能求第四个！');
},
speakCoprime() {
this.speak('互质关系很特殊：如果两个数互质，也就是最大公约数是1，那么它们的最小公倍数就等于两数的乘积。相邻自然数一定互质，两个不同的质数也一定互质。');
},
speakMultiple() {
this.speak('倍数关系最简单：如果A是B的倍数，那么它们的最大公约数就是较小的数B，最小公倍数就是较大的数A。因为大数已经包含了小数。');
},
speakScale() {
this.speak('同倍缩放是计算大数的神器：两个数同时乘以或除以同一个数，它们的最大公约数和最小公倍数也会同步变化。算大数时先缩小，算完再放大回去！');
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gsap.fromTo(box,
{opacity: 0, scale: 0},
{opacity: 1, scale: 1, duration: 0.5, delay}
);
return box;
};
const createOp = (text, delay) => {
const op = document.createElement('div');
op.className = 'operator';
op.textContent = text;
container.appendChild(op);
gsap.fromTo(op,
{opacity: 0},
{opacity: 1, duration: 0.3, delay}
);
return op;
};
// 第一行容器
const row1 = document.createElement('div');
row1.style.cssText = 'display: flex; justify-content: center; align-items: center; gap: 15px; flex-wrap: wrap;';
container.appendChild(row1);
// 临时将元素添加到row1
const oldAppend = container.appendChild;
container.appendChild = (el) => {
if (el.className === 'number-box' || el.className === 'operator') {
row1.appendChild(el);
} else {
oldAppend.call(container, el);
}
};
// A × B = 216
createBox('12', '#3B82F6', 0);
createOp('×', 0.6);
createBox('18', '#3B82F6', 1.2);
createOp('=', 1.8);
createBox('216', '#8B5CF6', 2.4);
// 恢复appendChild
container.appendChild = oldAppend;
// 3秒后显示质因数分解
setTimeout(() => {
const separator1 = document.createElement('div');
separator1.style.cssText = 'width: 100%; height: 1px; background: rgba(255,255,255,0.2); margin: 15px 0;';
container.appendChild(separator1);
gsap.fromTo(separator1, {opacity: 0}, {opacity: 1, duration: 0.3});
// 质因数分解说明
const factorSection = document.createElement('div');
factorSection.style.cssText = 'width: 100%; color: white; font-size: 12px; line-height: 1.6; margin: 10px 0;';
factorSection.innerHTML = `
<div style="display: flex; justify-content: space-around; gap: 10px; margin: 10px 0;">
<div style="flex: 1; text-align: center; background: rgba(59, 130, 246, 0.15); padding: 10px; border-radius: 8px; border: 1px solid #3B82F6;">
<div style="color: #60A5FA; font-weight: bold; margin-bottom: 5px; font-size: 11px;">12的分解</div>
<div style="font-size: 13px; font-weight: bold;">12 = 2² × 3</div>
</div>
<div style="flex: 1; text-align: center; background: rgba(59, 130, 246, 0.15); padding: 10px; border-radius: 8px; border: 1px solid #3B82F6;">
<div style="color: #60A5FA; font-weight: bold; margin-bottom: 5px; font-size: 11px;">18的分解</div>
<div style="font-size: 13px; font-weight: bold;">18 = 2 × 3²</div>
</div>
</div>
`;
container.appendChild(factorSection);
gsap.fromTo(factorSection, {opacity: 0, y: -10}, {opacity: 1, y: 0, duration: 0.5, delay: 0.3});
// 原理说明
setTimeout(() => {
const principle = document.createElement('div');
principle.style.cssText = 'width: 100%; background: rgba(16, 185, 129, 0.15); padding: 12px; border-radius: 8px; border: 1px solid #10B981; margin: 10px 0;';
principle.innerHTML = `
<div style="color: #34D399; font-weight: bold; margin-bottom: 8px; font-size: 12px;">📐 质因数视角</div>
<div style="font-size: 11px; line-height: 1.6; color: #CBD5E1;">
<strong style="color: #10B981;">GCD(12,18)</strong> = 2¹ × 3¹ = <strong>6</strong> （取最小指数）<br>
<strong style="color: #F59E0B;">LCM(12,18)</strong> = 2² × 3² = <strong>36</strong> （取最大指数）
</div>
`;
container.appendChild(principle);
gsap.fromTo(principle, {opacity: 0, y: -10}, {opacity: 1, y: 0, duration: 0.5});
}, 800);
setTimeout(() => {
const separator2 = document.createElement('div');
separator2.style.cssText = 'width: 100%; height: 1px; background: rgba(255,255,255,0.2); margin: 15px 0;';
container.appendChild(separator2);
gsap.fromTo(separator2, {opacity: 0}, {opacity: 1, duration: 0.3});
// 第二行容器
const row2 = document.createElement('div');
row2.style.cssText = 'display: flex; justify-content: center; align-items: center; gap: 15px; flex-wrap: wrap;';
container.appendChild(row2);
// 临时修改appendChild
const oldAppend2 = container.appendChild;
container.appendChild = (el) => {
if (el.className === 'number-box' || el.className === 'operator') {
row2.appendChild(el);
} else {
oldAppend2.call(container, el);
}
};
// GCD × LCM = 216
createBox('6', '#10B981', 0.5);
createOp('×', 1.1);
createBox('36', '#F59E0B', 1.7);
createOp('=', 2.3);
createBox('216', '#8B5CF6', 2.9);
// 恢复appendChild
container.appendChild = oldAppend2;
setTimeout(() => {
document.getElementById('product-text').textContent = '✓ 验证成功！12×18 = 6×36 = 216';
}, 3500);
}, 1500);
}, 3000);
document.getElementById('product-text').textContent = '例子：A=12，B=18，验证 A×B = GCD×LCM';
},
// 互质关系动画
animateCoprime() {
if (typeof gsap === 'undefined') return;
const container = document.getElementById('coprime-demo');
container.innerHTML = '';
container.style.flexDirection = 'column';
const createBox = (text, bg, delay) => {
const box = document.createElement('div');
box.className = 'number-box';
box.textContent = text;
box.style.background = bg;
container.appendChild(box);
gsap.fromTo(box,
{opacity: 0, scale: 0},
{opacity: 1, scale: 1, duration: 0.5, delay}
);
return box;
};
const createOp = (text, delay) => {
const op = document.createElement('div');
op.className = 'operator';
op.textContent = text;
container.appendChild(op);
gsap.fromTo(op,
{opacity: 0},
{opacity: 1, duration: 0.3, delay}
);
return op;
};
// 第一行容器
const row1 = document.createElement('div');
row1.style.cssText = 'display: flex; justify-content: center; align-items: center; gap: 15px; flex-wrap: wrap;';
container.appendChild(row1);
const oldAppend = container.appendChild;
container.appendChild = (el) => {
if (el.className === 'number-box' || el.className === 'operator') {
row1.appendChild(el);
} else {
oldAppend.call(container, el);
}
};
// 7 和 9 互质
createBox('7', '#3B82F6', 0);
createOp('和', 0.6);
createBox('9', '#3B82F6', 1.2);
createOp('互质', 1.8);
container.appendChild = oldAppend;
// 2.5秒后显示质因数分解
setTimeout(() => {
const separator1 = document.createElement('div');
separator1.style.cssText = 'width: 100%; height: 1px; background: rgba(255,255,255,0.2); margin: 15px 0;';
container.appendChild(separator1);
gsap.fromTo(separator1, {opacity: 0}, {opacity: 1, duration: 0.3});
// 质因数分解说明
const factorSection = document.createElement('div');
factorSection.style.cssText = 'width: 100%; color: white; font-size: 12px; line-height: 1.6; margin: 10px 0;';
factorSection.innerHTML = `
<div style="display: flex; justify-content: space-around; gap: 10px; margin: 10px 0;">
<div style="flex: 1; text-align: center; background: rgba(59, 130, 246, 0.15); padding: 10px; border-radius: 8px; border: 1px solid #3B82F6;">
<div style="color: #60A5FA; font-weight: bold; margin-bottom: 5px; font-size: 11px;">7的分解</div>
<div style="font-size: 13px; font-weight: bold;">7 = 7¹</div>
<div style="font-size: 10px; color: #93C5FD; margin-top: 3px;">（质数）</div>
</div>
<div style="flex: 1; text-align: center; background: rgba(59, 130, 246, 0.15); padding: 10px; border-radius: 8px; border: 1px solid #3B82F6;">
<div style="color: #60A5FA; font-weight: bold; margin-bottom: 5px; font-size: 11px;">9的分解</div>
<div style="font-size: 13px; font-weight: bold;">9 = 3²</div>
<div style="font-size: 10px; color: #93C5FD; margin-top: 3px;">（合数）</div>
</div>
</div>
`;
container.appendChild(factorSection);
gsap.fromTo(factorSection, {opacity: 0, y: -10}, {opacity: 1, y: 0, duration: 0.5, delay: 0.3});
// 互质原理说明
setTimeout(() => {
const principle = document.createElement('div');
principle.style.cssText = 'width: 100%; background: rgba(251, 191, 36, 0.15); padding: 12px; border-radius: 8px; border: 1px solid #F59E0B; margin: 10px 0;';
principle.innerHTML = `
<div style="color: #FCD34D; font-weight: bold; margin-bottom: 8px; font-size: 12px;">🤝 互质原理</div>
<div style="font-size: 11px; line-height: 1.6; color: #CBD5E1;">
7和9<strong style="color: #FCD34D;">没有公共质因数</strong><br>
所以 GCD(7,9) = 1，LCM(7,9) = 7 × 9
</div>
`;
container.appendChild(principle);
gsap.fromTo(principle, {opacity: 0, y: -10}, {opacity: 1, y: 0, duration: 0.5});
}, 800);
setTimeout(() => {
const separator2 = document.createElement('div');
separator2.style.cssText = 'width: 100%; height: 1px; background: rgba(255,255,255,0.2); margin: 15px 0;';
container.appendChild(separator2);
gsap.fromTo(separator2, {opacity: 0}, {opacity: 1, duration: 0.3});
const row2 = document.createElement('div');
row2.style.cssText = 'display: flex; justify-content: center; align-items: center; gap: 15px; flex-wrap: wrap;';
container.appendChild(row2);
const oldAppend2 = container.appendChild;
container.appendChild = (el) => {
if (el.className === 'number-box' || el.className === 'operator') {
row2.appendChild(el);
} else {
oldAppend2.call(container, el);
}
};
// 7 × 9 = 63
createBox('7', '#10B981', 0.5);
createOp('×', 1.1);
createBox('9', '#10B981', 1.7);
createOp('=', 2.3);
createBox('63', '#F59E0B', 2.9);
container.appendChild = oldAppend2;
setTimeout(() => {
document.getElementById('coprime-text').textContent = '✓ LCM(7, 9) = 7 × 9 = 63（互质直接相乘）';
}, 3500);
}, 1500);
}, 2500);
document.getElementById('coprime-text').textContent = '互质例子：7和9，GCD=1';
},
// 倍数关系动画
animateMultiple() {
if (typeof gsap === 'undefined') return;
const container = document.getElementById('multiple-demo');
container.innerHTML = '';
const createBox = (text, bg, delay, size = 60) => {
const box = document.createElement('div');
box.className = 'number-box';
box.textContent = text;
box.style.background = bg;
box.style.width = size + 'px';
box.style.height = size + 'px';
container.appendChild(box);
gsap.fromTo(box,
{opacity: 0, scale: 0},
{opacity: 1, scale: 1, duration: 0.5, delay}
);
return box;
};
const createOp = (text, delay) => {
const op = document.createElement('div');
op.className = 'operator';
op.textContent = text;
container.appendChild(op);
gsap.fromTo(op,
{opacity: 0},
{opacity: 1, duration: 0.3, delay}
);
return op;
};
// 6 × 3 = 18
createBox('6', '#3B82F6', 0, 50);
createOp('×3=', 0.6);
createBox('18', '#8B5CF6', 1.2, 80);
setTimeout(() => {
const text = document.createElement('div');
text.style.cssText = 'color: #FCD34D; font-size: 16px; font-weight: bold; text-align: center; width: 100%; margin-top: 20px;';
text.innerHTML = 'GCD = 6（小的）<br>LCM = 18（大的）';
container.appendChild(text);
gsap.fromTo(text, {opacity: 0}, {opacity: 1, duration: 0.5});
document.getElementById('multiple-text').textContent = '18是6的3倍，所以GCD=6，LCM=18';
}, 2000);
document.getElementById('multiple-text').textContent = '倍数关系：18 = 6 × 3';
},
// 同倍缩放动画
animateScale() {
if (typeof gsap === 'undefined') return;
const container = document.getElementById('scale-demo');
container.innerHTML = '';
const steps = [
{eq: '原问题：GCD(1200, 1600) = ?', exp: '大数不好算'},
{eq: '提取公因数：1200 = 12 × 100，1600 = 16 × 100', exp: '发现都是100的倍数'},
{eq: '缩小100倍：GCD(12, 16) = ?', exp: '先算小数'},
{eq: 'GCD(12, 16) = 4', exp: '计算结果'},
{eq: '放大100倍：4 × 100 = <span class="calc-highlight">400</span>', exp: '还原答案'}
];
steps.forEach((step, i) => {
const div = document.createElement('div');
div.className = 'calc-step';
div.innerHTML = `
<div class="calc-equation">${step.eq}</div>
<div class="calc-explanation">${step.exp}</div>
`;
container.appendChild(div);
gsap.fromTo(div,
{opacity: 0, y: 20},
{
opacity: 1,
y: 0,
duration: 0.5,
delay: i * 1.5,
onStart: () => {
if (i > 0) {
for (let j = 0; j < i; j++) {
container.children[j].classList.remove('active');
}
}
div.classList.add('active');
document.getElementById('scale-text').textContent = `第${i+1}步：${step.exp}`;
}
}
);
});
document.getElementById('scale-text').textContent = '点击按钮开始演示';
this.$nextTick(() => this.renderMath());
},
// 例题切换
switchExample(index) {
this.exampleIndex = index;
this.$nextTick(() => this.renderMath());
},
nextExampleStep(index) {
if (this.exampleSteps[index] < this.classicExamples[index].steps.length - 1) {
this.exampleSteps[index]++;
this.$nextTick(() => this.renderMath());
}
},
prevExampleStep(index) {
if (this.exampleSteps[index] > 0) {
this.exampleSteps[index]--;
}
},
resetExampleStep(index) {
this.exampleSteps[index] = 0;
},
// 练习题
checkPracticeAnswer() {
if (this.practiceAnswered) return;
const correctAnswer = this.practiceQuestions[this.currentPracticeIndex].answer.toLowerCase().trim();
const userAnswer = this.userAnswer.toLowerCase().trim().replace(/\s+/g, '');
const correct = userAnswer === correctAnswer ||
userAnswer === correctAnswer.replace(/\*/g, '') ||
userAnswer === correctAnswer.replace('*', 'x');
this.practiceAnswered = true;
this.isPracticeCorrect = correct;
if (correct) {
this.practiceScore += 10;
}
this.$nextTick(() => this.renderMath());
},
nextPractice() {
if (this.currentPracticeIndex < 9) {
this.currentPracticeIndex++;
this.practiceAnswered = false;
this.userAnswer = '';
this.$nextTick(() => this.renderMath());
} else {
this.practiceDone = true;
}
},
// 杯赛真题
checkOlympiadAnswer() {
if (this.olympiadAnswered) return;
const correctAnswer = this.olympiadQuestions[this.currentOlympiadIndex].answer.toLowerCase().trim();
const userAnswer = this.userOlympiadAnswer.toLowerCase().trim().replace(/\s+/g, '');
const correct = userAnswer === correctAnswer ||
userAnswer === correctAnswer.replace(',', '或') ||
userAnswer === correctAnswer.replace(',', ' ');
this.olympiadAnswered = true;
this.isOlympiadCorrect = correct;
if (correct) {
this.olympiadScore += 10;
}
this.$nextTick(() => this.renderMath());
},
nextOlympiad() {
if (this.currentOlympiadIndex < 9) {
this.currentOlympiadIndex++;
this.olympiadAnswered = false;
this.userOlympiadAnswer = '';
this.$nextTick(() => this.renderMath());
} else {
this.olympiadDone = true;
}
}
},
mounted() {
this.renderMath();
}
}).mount('#app');
</script>
</body>
</html>💡 这段代码完全由 claude 生成。
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