The Magic Number 245: Prime Factorisation and Surds | 数字245：质因数分解与根式化简

📚 The Magic Number 245: Prime Factorisation and Surds | 数字245：质因数分解与根式化简

Why start a revision article with a random number like 245? Because this humble three-digit integer hides several essential Edexcel IGCSE Mathematics topics: prime factorisation, square roots, surds, and fractional indices. By working through the properties of 245, you will strengthen skills that appear regularly in Paper 1 and Paper 2.

为什么用245这个看似随意的数字来开启一篇复习文章？因为这个不起眼的三位整数背后藏着EDEXCEL IGCSE数学的若干核心专题：质因数分解、平方根、根式以及分数指数。通过挖掘245的性质，你能够巩固在Paper 1和Paper 2中反复出现的技能。

1. What Is Prime Factorisation? | 什么是质因数分解？

Every integer greater than 1 can be written uniquely as a product of prime numbers. This is called the Fundamental Theorem of Arithmetic. For example, \(12 = 2^2 \times 3\) but we do not use LaTeX here, so we write: 12 = 2² × 3.

每个大于1的整数都可以唯一地写成素数的乘积，这就是算术基本定理。例如，12 = 2² × 3。

A prime number is a positive integer greater than 1 that has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, …

素数是指大于1且恰好只有两个不同因数（1和它本身）的正整数。前几个素数是2、3、5、7、11、13、17、19……

Prime factorisation is a method of breaking a number into its prime building blocks. It is the foundation for finding HCF (highest common factor), LCM (lowest common multiple), and for simplifying surds.

质因数分解是将一个数拆分成素数构件的方法。它是求最大公因数（HCF）、最小公倍数（LCM）以及化简根式的基础。

2. Factorising 245 | 分解245

Let us apply this method to 245. First, note that 245 is not even, so it is not divisible by 2. Does it end in 5? Yes, so it is divisible by 5. Divide: 245 ÷ 5 = 49.

让我们将此方法应用于245。首先，245不是偶数，因此不能被2整除。它以5结尾，因此能被5整除。计算：245 ÷ 5 = 49。

Now factorise 49. Since 49 = 7 × 7 and 7 is prime, the prime factorisation of 245 is:

现在分解49。因为49 = 7 × 7，且7是素数，所以245的质因数分解为：

245 = 5 × 7 × 7 = 5 × 7²

We can check: 5 × 49 = 245. This compact form, using an index (power), tells us the complete prime structure of 245.

我们可以检验：5 × 49 = 245。这种使用指数（幂）的紧凑形式揭示了245完整的素数结构。

If you use a factor tree, start with any factor pair, for example 245 = 5 × 49, then split 49 = 7 × 7. The leaves are all primes: 5, 7, 7.

如果你使用因子树，可以从任意一对因数开始，例如245 = 5 × 49，然后将49分成7 × 7。所有末端都是素数：5、7、7。

3. Significance of Prime Factors | 质因数的意义

Knowing that 245 = 5 × 7² allows us to solve many problems without recalculating from scratch.

知道245 = 5 × 7²，使我们可以不从头重新计算就能解决许多问题。

Factors: The factors of 245 are 1, 5, 7, 35, 49, 245. Notice that 35 = 5 × 7 and 49 = 7².
因数：245的因数是1、5、7、35、49、245。注意35 = 5 × 7，49 = 7²。
Number of factors: If the prime factorisation is \(p^a q^b\), the total number of factors is \((a+1)(b+1)\). Here \(5^1 \times 7^2\) gives (1+1)(2+1) = 2 × 3 = 6 factors. We listed them above.
因数个数：如果质因数分解是pᵃ qᵇ，那么总因数个数为(a+1)(b+1)。这里5¹ × 7²，得到(1+1)(2+1) = 2 × 3 = 6个因数。上面我们已经全部列出。
Square factor: 245 contains a perfect square factor, 49. This is the key to simplifying the square root.
平方因数：245含有一个完全平方因数49。这是化简平方根的关键。

In Edexcel IGCSE, questions might ask you to find the prime factorisation of a product, or to write a number as a product of primes. Always make sure your final answer uses index notation correctly.

在Edexcel IGCSE中，题目可能要求你求某个数的质因数分解，或将一个数写成素数乘积的形式。务必确保最终答案正确使用指数记号。

4. Square Roots and Surds | 平方根与根式

The square root of a number \(n\) is a value that, when multiplied by itself, gives \(n\). For example, \(\sqrt{49}=7\) because \(7^2=49\).

一个数n的平方根是这样一个值，它自乘后等于n。例如，√49 = 7，因为7² = 49。

But what about \(\sqrt{245}\)? There is no whole number whose square is 245. Such numbers produce irrational results, and when written with a radical sign, they are called surds (or radicals). A surd is an irrational number that can be expressed exactly as the square root of an integer that is not a perfect square.

但√245呢？没有任何整数的平方等于245。这类数产生无理数结果，当用根号表示时，称为根式（surds）。根式是一个无理数，可以精确地表示为一个非完全平方整数的平方根。

For IGCSE, you do not need to calculate a long decimal; you must leave the answer in surd form unless told otherwise. Therefore, simplifying surds is a vital skill.

对于IGCSE，你无需计算长小数；除非另有说明，你必须将答案保留为根式形式。因此，化简根式是一项至关重要的技能。

5. Simplifying √245 | 化简√245

To simplify a square root, look for the largest perfect square factor of the number under the root. From the prime factorisation 245 = 5 × 7², we see that 7² is a perfect square.

要化简平方根，需要寻找根号下这个数的最大完全平方因数。从质因数分解245 = 5 × 7²中，我们看到7²是完全平方数。

Write the root as a product:

将根号写成乘积：

√245 = √(5 × 7²) = √(7² × 5) = √(7²) × √5 = 7√5

Since 5 is prime and has no square factors, \(7\sqrt{5}\) is the fully simplified form.

因为5是素数且没有平方因数，所以7√5就是最简形式。

Remember to check that you have taken out the largest possible square factor. If you instead wrote √245 = √(5 × 49), you still get the same result because 49 is the largest square factor. If you miss it and write √245 = √5 × √49, that is fine too. The key is to recognise the perfect square.

记住要检查你是否提取了最大的平方因数。如果你改为写成√245 = √(5 × 49)，结果依然相同，因为49是最大的平方因数。如果你漏掉它而写√245 = √5 × √49，那也没问题。关键是识别完全平方数。

Common mistake: students sometimes write \(\sqrt{245} = \sqrt{5} \times \sqrt{7}\) which is wrong because 245 = 5 × 49, not 5 × 7. Always double-check your factorisation.

常见错误：学生有时会写√245 = √5 × √7，这是错误的，因为245 = 5 × 49，而不是5 × 7。务必仔细检查你的分解。

6. Fractional Indices and Roots | 分数指数与根式

In Edexcel IGCSE, you must be comfortable with fractional indices. Recall that \(a^{1/2} = \sqrt{a}\) and \(a^{1/3} = \sqrt[3]{a}\). More generally, \(a^{m/n} = (\sqrt[n]{a})^m\).

在Edexcel IGCSE中，你必须熟悉分数指数。回忆：a^(1/2) = √a，a^(1/3) = ∛a。更一般地，a^(m/n) = (n次根号下a)的m次方。

Using 245, we can write:

利用245，我们可以写出：

245^{1/2} = √245 = 7√5

Also, \(245^{3/2}\) means \((245^{1/2})^3 = (7\sqrt{5})^3 = 343 \times 5\sqrt{5} = 1715\sqrt{5}\). Let’s check: \(7^3 = 343\), and \((\sqrt{5})^3 = 5\sqrt{5}\), so multiplying gives \(343 \times 5\sqrt{5} = 1715\sqrt{5}\).

同时，245^(3/2) 表示 (245^(1/2))³ = (7√5)³ = 343 × 5√5 = 1715√5。我们检验：7³ = 343，(√5)³ = 5√5，相乘得到343 × 5√5 = 1715√5。

Such calculations combine indices, surds, and arithmetic. Make sure you know the laws of indices:

这类计算结合了指数、根式和算术。确保你掌握指数法则：

\(a^m \times a^n = a^{m+n}\) → aᵐ × aⁿ = aᵐ⁺ⁿ
\(a^m ÷ a^n = a^{m-n}\) → aᵐ ÷ aⁿ = aᵐ⁻ⁿ
\((a^m)^n = a^{mn}\) → (aᵐ)ⁿ = aᵐⁿ
\(a^{-m} = 1/a^m\) → a⁻ᵐ = 1/aᵐ

In the Edexcel IGCSE calculator paper, you might see a question like “Simplify \(245^{1/2}\)” without a calculator; the answer is \(7\sqrt{5}\).

在Edexcel IGCSE可用计算器的试卷中，你可能会看到类似“化简245^(1/2)”的题目，答案就是7√5。

7. Applications of 245 | 245的应用

Where might the number 245 appear in exam problems?

数字245在考试题目中可能出现在哪里？

Area of a square: If the area of a square is 245 cm², its side length is \(\sqrt{245} = 7\sqrt{5}\) cm. This may be left in surd form.
面积问题：如果正方形的面积是245平方厘米，其边长就是√245 = 7√5厘米。答案可以保留根式形式。
Volume problems: A cuboid with dimensions 5, 7, 7 has volume \(5 × 7 × 7 = 245\). This ties to prime factors.
体积问题：一个长、宽、高分别为5、7、7的长方体体积为5 × 7 × 7 = 245。这与质因数关联。
Pythagoras: In a right-angled triangle, if one leg is \(\sqrt{5}\) and the other leg is \(7\sqrt{5}\), then the hypotenuse squared is \((\sqrt{5})^2 + (7\sqrt{5})^2 = 5 + 245 = 250\), so the hypotenuse is \(\sqrt{250} = 5\sqrt{10}\). Here 245 appears as a square.
勾股定理：在直角三角形中，如果一条直角边为√5，另一条为7√5，那么斜边的平方是(√5)² + (7√5)² = 5 + 245 = 250，因此斜边为√250 = 5√10。这里245作为一个平方出现。
HCF and LCM: Compare 245 with another number, say 105 = 3 × 5 × 7. The HCF of 245 and 105 is 5 × 7 = 35. The LCM is \(3 × 5 × 7² = 735\).
最大公因数与最小公倍数：将245与另一个数（如105 = 3 × 5 × 7）比较。245和105的HCF是5 × 7 = 35。LCM是3 × 5 × 7² = 735。

These problems show that prime factorisation is not an isolated topic; it connects to geometry, number theory, and algebra.

这些问题表明质因数分解并非孤立主题；它与几何、数论和代数相互联系。

8. Common Exam Tips and Mistakes | 常见考点与易错点

Let us review some pitfalls and exam tips related to numbers like 245.

让我们回顾与245这类数相关的一些陷阱和考点提示。

Mistake / 易错点	Correction / 正确做法
Writing 245 = 5 × 7 × 5 (incorrectly splitting 49 as 7 × 5)	49 = 7 × 7, so 245 = 5 × 7 × 7
Leaving √245 without simplifying	Always factor out perfect squares: √245 = 7√5
Confusing \(245^{1/2}\) with \(1/(245^2)\)	\(245^{1/2}\) is the square root, not the reciprocal of the square

Exam tip: When simplifying a square root, first write the number as a product of a perfect square and another integer. The perfect square should be the largest one. For 245, the largest perfect square factor is 49 (not 1).

考试提示：化简平方根时，先把数写成完全平方数与另一个整数的乘积。这个完全平方数应是最大的。对于245，最大的完全平方因数是49（不是1）。

Another tip: If the question asks for the answer in the form \(a\sqrt{b}\) where \(b\) is a positive integer and \(a\) is an integer, state your final answer clearly: \(7\sqrt{5}\), so \(a=7\), \(b=5\).

另一个提示：如果题目要求把答案写成\(a\sqrt{b}\)的形式，其中\(b\)为正整数，\(a\)为整数，请清晰写出最终答案：\(7\sqrt{5}\)，因此a=7，b=5。

9. Practice Questions | 练习题

Try these on your own to master the concepts.

尝试自己完成以下题目以掌握这些概念。

Express 245 as a product of its prime factors.

将245表示为质因数的乘积。
Simplify \(\sqrt{245}\).

化简√245。
Work out \(245^{3/2}\) and give your answer in the form \(a\sqrt{b}\).

计算245^(3/2)，并将答案写成a√b的形式。
The area of a rectangle is 245 cm² and its width is \(\sqrt{5}\) cm. Find the length in the form \(p\sqrt{q}\).

一个矩形的面积是245平方厘米，宽是√5厘米。求长度，用p√q的形式表示。

Answers: 1) \(5 × 7^2\) or 5 × 7²; 2) \(7\sqrt{5}\); 3) \(1715\sqrt{5}\); 4) Length = 245/√5 = 245√5/5 = 49√5, so \(p=49\), \(q=5\).

答案：1) 5 × 7²；2) 7√5；3) 1715√5；4) 长度 = 245/√5 = 245√5/5 = 49√5，所以p=49，q=5。

10. Summary | 总结

The number 245 is more than just a three-digit integer. Through it, we practised prime factorisation (5 × 7²), square root simplification (7√5), and fractional indices \((245^{1/2} = 7√5, 245^{3/2} = 1715√5)\). We also connected these skills to geometry, HCF/LCM, and common exam pitfalls.

245这个数字不仅仅是一个三位整数。通过它，我们练习了质因数分解（5 × 7²）、平方根化简（7√5）以及分数指数（245^(1/2) = 7√5，245^(3/2) = 1715√5）。我们还将这些技能联系到了几何、HCF/LCM以及常见考试陷阱。

Remember these key points for your Edexcel IGCSE Mathematics exam:

记住以下Edexcel IGCSE数学考试的关键点：

Always break numbers into primes before dealing with roots or fractions.
在处理根式或分数之前，务必先把数分解为素数。
Look for the largest perfect square factor when simplifying surds.
化简根式时寻找最大的完全平方因数。
Use the laws of indices accurately for fractional powers.
精确使用指数法则处理分数幂。

Mastering these techniques will not only help you answer questions about 245 but also any similar surd and index problems that appear in the exam.

掌握这些技巧不仅会帮助你回答关于245的问题，也会帮助你应对考试中出现的任何类似的根式和指数问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

The Magic Number 245: Prime Factorisation and Surds | 数字245：质因数分解与根式化简