📚 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 — torise, 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。
n (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。
s 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
ver, 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：
1441/2 = √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
× 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? s 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
d 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 |
air 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。
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
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：
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。
Divisible by 8 because "144" is 8 × 18.
能被8整除，因为144 = 8 × 18。
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。
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) = 122+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) = 122+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：
144⁻¹ = 1/144
144⁻¹ = 1/144
144⁻¹ᐟ² = 1/√144 = 1/12
144⁻¹ᐟ² = 1/√144 = 1/12
1443/2 = (√144)³ = 12³ = 1 728
1443/2 = (√144)³ = 12³ = 1728
Questions like "evaluate 1443/2" 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1443/2”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
f 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
everse 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。
. 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。
² = 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：
36/144 = 1/4 = 25%
36/144 = 1/4 = 25%
48/144 = 1/3 ≈ 33.3%
48/144 = 1/3 ≈ 33.3%
72/144 = 1/2 = 50%
72/144 = 1/2 = 50%
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：
√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。
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。
er 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。
1443/2 equals 1 728, not 144³ or 144 × 3 ÷ 2. Convert to root-first form: (√144)³.
1443/2等于1728，而不是144³或144 × 3 ÷ 2。应化为先开根的形式：(√144)³。
k 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：
Q1. Evaluate √144 + √36.
题1. 计算√144 + √36。
Q2. Write 144 as a product of prime factors.
题2. 将144写成质因数的乘积。
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。
Q4. How many positive factors does 144 have?
题4. 144有多少个正因数？
Q5. Simplify the ratio 144 :
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
广告 / Sponsored预售 英国CGP教材 GCSE/IGCSE Catch-Up Essentials:Edexcel Maths Hig…数学 备考教辅 · 正版现货 · 与本文内容相关查看详情 / View on Taobao参考价 ¥209.0更多咨询请联系16621398022（同微信）
备考资讯每日更新，关注不错过考纲变化 · 科目分析 · 学习方法
在微信里长按二维码识别关注
或截图后用「扫一扫 → 相册」选择本图公众号：tutorhao