📚 Mastering 144: Squares, Roots and Prime Factorisation | 掌握144：平方、根与质因数分解
The number 144 may look like an ordinary three-digit integer, but it holds a special place in IGCSE Edexcel Mathematics. As the square of 12, a highly composite number, and a versatile example for primes, powers and geometry, 144 appears again and again across the syllabus. Mastering this single number gives you a powerful toolkit for handling squares, square roots, factorisation and problem solving in your exams.
数字144看似只是一个普通的三位整数，但在IGCSE Edexcel数学中却占有特殊地位。作为12的平方、一个高合成数，以及质数、幂与几何中的多用途例子，144在考纲中反复出现。彻底掌握这一个数字，能为你应对考试中的平方、平方根、分解因数与问题求解提供强有力的工具包。
1. Why 144 Matters | 为什么144重要
144 is a perfect square because 12 × 12 = 144. It is also called a “gross” — a dozen dozen — and it is one of the most frequently used numbers in exercises on squares, roots, factors and indices. Because 144 is relatively large yet easy to factorise, examiners love to use it as a test of whether you truly understand the underlying number structure.
144是一个完全平方数，因为12 × 12 = 144。它也被称为“一罗”——即一打打——是平方、根、因数与指数练习中最常出现的数字之一。由于144相对较大却又容易分解，命题者非常喜欢用它来检验你是否真正理解底层的数结构。
In the Edexcel IGCSE syllabus, you will meet 144 in topics ranging from Number and Algebra to Mensuration. Knowing its prime factorisation (2⁴ × 3²) and its square root (12) will save you valuable time in non-calculator papers.
在Edexcel IGCSE考纲中，从“数与代数”到“求积法”等专题都会遇到144。熟悉它的质因数分解（2⁴ × 3²）和平方根（12），能让你在不使用计算器的试卷中节省宝贵时间。
2. Squaring — The Origin of 144 | 平方——144的来源
To square a number means to multiply it by itself. The square of 12 is written with a superscript 2:
一个数的平方意味着将它自乘一次。12的平方用上标2表示：
12² = 12 × 12 = 144
The superscript “2” tells us that the base (12) is used as a factor twice. Squaring is a fundamental operation in the IGCSE course, and 144 is one of the cleanest examples because it has no awkward decimals.
上标“2”告诉我们底数（12）作为因数被使用了两次。平方是IGCSE课程中的基础运算，而144是最干净利落的例子之一，因为它不涉及任何烦人的小数。
Other squares that frequently appear alongside 144 include 11² = 121, 13² = 169 and 14² = 196. Notice that 12² sits almost exactly between 11² and 13², making 144 a useful anchor point for estimating other square roots.
与144经常一同出现的其他平方数包括11² = 121、13² = 169和14² = 196。注意12²恰好位于11²与13²之间，使144成为估算其他平方根时非常有用的锚点。
3. Square Roots — Going Back to 12 | 平方根——回到12
The inverse operation of squaring is taking the square root. The symbol √ asks: “which positive number, when multiplied by itself, gives this value?”
平方的逆运算是开平方。符号√询问的是：“哪一个正数自乘后得到这个值？”
√144 = 12 because 12 × 12 = 144
In exact form, √144 is a rational number because 144 is a perfect square. The principal square root is always non-negative, so √144 = 12, not −12. However, when solving an equation like x² = 144, both solutions x = +12 and x = −12 are valid.
在精确形式中，√144是一个有理数，因为144是完全平方数。主平方根永远是非负的，因此√144 = 12，而不是−12。但在解方程x² = 144时，x = +12和x = −12两个解都成立。
You can also write the square root using a fractional index:
你也可以用分数指数表示平方根：
144^(1/2) = √144 = 12
This link between surds and indices is a common exam question. Remember that 144 is one of the few squares whose root is a simple two-digit integer, so it is an excellent mental arithmetic benchmark.
根式与指数之间的这种联系是常见的考题。记住，144是少数几个平方根为简洁两位整数的平方数之一，因此它是心算的理想基准。
4. Prime Factorisation of 144 | 144的质因数分解
Prime factorisation means writing a number as a product of prime numbers only. For 144, we can repeatedly divide by the smallest prime:
质因数分解指的是将一个数仅写成质数的乘积。对于144，我们可以用最小的质数反复相除：
144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2⁴ × 3²
A division diagram (factor tree) is often the quickest method: split 144 into 12 × 12, then split each 12 into 2 × 2 × 3. Collecting the prime factors gives 2 × 2 × 2 × 2 × 3 × 3, which we write as 2⁴ × 3².
因式分解树（factor tree）通常是最快的方法：将144拆成12 × 12，再将每个12拆成2 × 2 × 3。收集所有质因数得到2 × 2 × 2 × 2 × 3 × 3，即写成2⁴ × 3²。
Why does this matter? The prime factor form explains everything about 144: its square root, its divisors, and whether it is a perfect square. Since every exponent in 2⁴ × 3² is even, 144 must be a perfect square.
为什么这很重要？质因数形式解释了关于144的一切：它的平方根、它的因数，以及它是否为完全平方数。因为2⁴ × 3²中的每个指数均为偶数，144必然是完全平方数。
5. Factor Pairs and Number of Divisors | 因数对与因数个数
Using the index form 2⁴ × 3², we can find the total number of positive divisors. The rule is: add 1 to each exponent and multiply.
利用指数形式2⁴ × 3²，我们可以求出正因数的总数。规则是：每个指数加1再相乘。
(4 + 1) × (2 + 1) = 5 × 3 = 15 divisors
The complete list of factor pairs is shown below:
完整的因数对如下表所示：
| 1 × 144 | 2 × 72 | 3 × 48 | 4 × 36 |
| 6 × 24 | 8 × 18 | 9 × 16 | 12 × 12 |
Notice that the factor pair 12 × 12 is special: it corresponds to the square root. Any perfect square always has a “middle” factor pair where both numbers are equal.
注意因数对12 × 12很特殊：它对应平方根。任何完全平方数总有一个“中间”因数对，其中两个数相等。
The sum of all divisors of 144 is (2⁵ − 1) ÷ 1 × (3³ − 1) ÷ 2 = 31 × 13 = 403. This type of divisor problem occasionally appears in higher-level IGCSE questions, so it is worth recognising the pattern.
144所有因数之和为(2⁵ − 1) ÷ 1 × (3³ − 1) ÷ 2 = 31 × 13 = 403。这类因数求和问题偶尔出现在IGCSE较难的题目中，值得识别其规律。
6. Multiples and Divisibility | 倍数与整除性
Because 144 = 2⁴ × 3², it is divisible by every number of the form 2ᵃ × 3ᵇ where a is from 0 to 4 and b is from 0 to 2. In simple terms, 144 is a multiple of 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 144.
因为144 = 2⁴ × 3²，它能被所有形如2ᵃ × 3ᵇ的数整除，其中a可取0到4，b可取0到2。简单来说，144是1、2、3、4、6、8、9、12、16、18、24、36、48、72和144的倍数。
Divisibility tests make working with 144 quick without a calculator:
整除性检验使你在没有计算器的情况下也能快速处理144：
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Divisible by 4 because the last two digits “44” form a multiple of 4.
能被4整除，因为末两位“44”是4的倍数。
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Divisible by 8 because “144” is 8 × 18.
能被8整除，因为144 = 8 × 18。
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Divisible by 9 because 1 + 4 + 4 = 9, and 9 is divisible by 9.
能被9整除，因为数位和1 + 4 + 4 = 9，而9能被9整除。
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Divisible by 3 because the digit sum 9 is a multiple of 3.
能被3整除，因为数位和9是3的倍数。
These tests are core IGCSE skills, and 144 is the perfect number to practise them on.
这些检验方法是IGCSE的核心技能，而144正是练习它们的最佳数字。
7. Indices and Power Laws | 指数与幂法则
Since 144 = 12², we can combine it with the laws of indices. For example:
由于144 = 12²，我们可以将其与指数法则结合。例如：
144 × 144 = 12² × 12² = 12⁴ = 20 736
When multiplying powers with the same base, we add the exponents: (12²) × (12²) = 12^(2+2) = 12⁴. When raising a power to a power, we multiply the exponents:
当同底数幂相乘时，指数相加：(12²) × (12²) = 12^(2+2) = 12⁴。当幂再乘方时，指数相乘：
(12²)³ = 12⁶ = 2 985 984
Negative and fractional indices also come into play:
负指数与分数指数同样适用：
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144⁻¹ = 1/144
144⁻¹ = 1/144
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144⁻¹ᐟ² = 1/√144 = 1/12
144⁻¹ᐟ² = 1/√144 = 1/12
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144^(3/2) = (√144)³ = 12³ = 1 728
144^(3/2) = (√144)³ = 12³ = 1728
Questions like “evaluate 144^(3/2)” are regular Edexcel exam items. Always take the root first when the fraction has a small denominator — it keeps the numbers manageable.
像“计算144^(3/2)”这样的题目是Edexcel考试的常规题型。当分数指数分母较小时，务必先开方——这样可以使数值保持在容易处理的范围内。
8. Geometry — Area of a Square | 几何——正方形的面积
In mensuration, the area of a square is given by side × side = side². A square with side length 12 cm therefore has:
在求积法中，正方形的面积等于边长 × 边长 = 边长²。因此，边长为12 cm的正方形面积为：
A = 12² = 144 cm²
Conversely, if a square has area 144 m², its side length is √144 = 12 m. This reverse process is a classic application of square roots in the Geometry and Mensuration topic.
反过来，如果一个正方形的面积为144 m²，其边长为√144 = 12 m。这一逆向过程是“几何与求积法”专题中平方根的经典应用。
The perimeter would then be 4 × 12 = 48 m. A common exam trap is to answer only with the side length when the question asks for the perimeter — always read carefully.
此时周长为4 × 12 = 48 m。一个常见的考试陷阱是题目问周长时只回答边长——务必仔细审题。
Beyond squares, 144 appears in circle problems: if the radius of a circle is 12 cm, the area is π × 12² = 144π cm². Leaving the answer in terms of π shows exact working and is often preferred.
除正方形外，144也出现在圆的问题中：若圆的半径为12 cm，则面积为π × 12² = 144π cm²。将答案保留为含π的形式体现了精确计算，且往往是更受青睐的写法。
9. Fractions, Ratios and Percentages | 分数、比率与百分比
Because 144 is a convenient “whole” total, it frequently appears as a denominator or baseline in fractions, ratios and percentages.
因为144是一个方便的“整体”总数，它经常作为分数、比率和百分比中的分母或基准。
Some key simplifications to remember:
以下关键化简值得记住：
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36/144 = 1/4 = 25%
36/144 = 1/4 = 25%
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48/144 = 1/3 ≈ 33.3%
48/144 = 1/3 ≈ 33.3%
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72/144 = 1/2 = 50%
72/144 = 1/2 = 50%
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Ratio 144 : 96 simplifies to 3 : 2 (divide both by 48).
比率144 : 96化简为3 : 2（两边同除以48）。
When a question gives you a large total like 144, always ask whether dividing by 12, 16 or 24 reveals a simpler fraction. This makes percentage and ratio problems dramatically faster in non-calculator exams.
当题目给出144这样较大的总数时，务必思考是否能除以12、16或24来得到一个更简单的分数。这能在不使用计算器的考试中大幅加快百分比与比率题的速度。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Even strong students lose marks on 144-based questions due to avoidable errors. Here are the most important points to remember:
即使是优秀的学生也会在与144相关的题目上因可避免的错误而失分。以下是需要记住的最重要的几点：
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√144 = 12 only. Do not write ±12 unless you are solving an equation such as x² = 144.
√144只等于12。除非是在解x² = 144这样的方程，否则不要写成±12。
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Prime factorisation of 144 is 2⁴ × 3². Do not swap the exponents to 2² × 3⁴ — that equals 324, not 144.
144的质因数分解是2⁴ × 3²。不要将指数交换为2² × 3⁴——那等于324，而不是144。
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When finding the number of divisors, use (4+1)(2+1) = 15, not 4 × 2 = 8. You must add 1 to each exponent first.
求因数个数时，应使用(4+1)(2+1) = 15，而不是4 × 2 = 8。必须先给每个指数加1。
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144^(3/2) equals 1 728, not 144³ or 144 × 3 ÷ 2. Convert to root-first form: (√144)³.
144^(3/2)等于1728，而不是144³或144 × 3 ÷ 2。应化为先开根的形式：(√144)³。
Always check whether your answer is sensible. If you get √144 = 14, you have misremembered the square — 14² = 196, not 144.
始终检查答案是否合理。如果你算出√144 = 14，那说明你记错了平方数——14² = 196，而不是144。
11. Practice Problem Set | 练习题库
Apply everything you have learned with this quick set of exam-style questions:
用下面这组考试风格的快速习题来应用你所学的内容：
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Q1. Evaluate √144 + √36.
题1. 计算√144 + √36。
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Q2. Write 144 as a product of prime factors.
题2. 将144写成质因数的乘积。
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Q3. The area of a square is 144 cm². Find its perimeter.
题3. 一个正方形的面积为144 cm²。求它的周长。
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Q4. How many positive factors does 144 have?
题4. 144有多少个正因数？
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Q5. Simplify the ratio 144 :
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