Mastering IGCSE Maths: A Study of 248 | IGCSE数学精通：数字248研究

📚 Mastering IGCSE Maths: A Study of 248 | IGCSE数学精通：数字248研究

In this Edexcel IGCSE Mathematics revision article, we investigate the number 248. Although it may look like an ordinary three-digit integer, 248 is an ideal example for revising fundamental number skills: place value, prime factorisation, factors, multiples, HCF and LCM, powers, standard form, rounding, divisibility and problem solving.

在本文中，我们将以数字248为研究对象。虽然它看起来只是一个普通的三位数，但它非常适合用来复习Edexcel IGCSE数学中基础“数”的技能：位值、质因数分解、因数、倍数、最大公因数(HCF)、最小公倍数(LCM)、幂、科学计数法、四舍五入、整除规则以及实际问题解决。

1. Place Value and Digits | 位值与数字

Every whole number is built from place value. In the number 248, the digit 2 is in the hundreds place, so its value is 2 × 100 = 200; the digit 4 is in the tens place, so its value is 4 × 10 = 40; the digit 8 is in the units place, so its value is 8.

每个整数都由位值构成。在数字248中，数字2位于百位，其值为2×100=200；数字4位于十位，其值为4×10=40；数字8位于个位，其值为8。

Thus 248 can be written in expanded form as 200 + 40 + 8. This is the basis of the base-10 number system and helps when rounding, estimating or converting to standard form.

因此248可以写成展开式200+40+8。这是十进制数系的基础，也便于后续进行四舍五入、估算或转换成科学计数法。

2. Prime Factorisation | 质因数分解

A prime number is a positive integer greater than 1 with exactly two factors: 1 and itself. Prime factorisation means writing a number as a product of primes. Using a short division method, we divide by primes until the final quotient is 1:

质数是指大于1且只有1和自身两个因数的正整数。质因数分解就是把一个数写成质数的乘积。用短除法逐次除以质数，直到最后的商为1：

248 ÷ 2 = 124, 124 ÷ 2 = 62, 62 ÷ 2 = 31, 31 ÷ 31 = 1

Since 2 and 31 are primes, the prime factorisation of 248 is:

因为2和31都是质数，所以248的质因数分解为：

248 = 2 × 2 × 2 × 31 = 2³ × 31

The prime factors of 248 are therefore 2 and 31. Notice that 2 appears three times, so we write it as 2³.

因此248的质因数是2和31。注意2出现了三次，所以我们写成2³。

3. Factors and Multiples | 因数与倍数

A factor of 248 is a whole number that divides 248 without leaving a remainder. From the prime factorisation 2³ × 31, we can generate all factor pairs:

248的因数是指能整除248且没有余数的整数。从质因数分解2³×31，我们可以得到所有因数对：

1 × 248
2 × 124
4 × 62
8 × 31

So the complete list of positive factors is 1, 2, 4, 8, 31, 62, 124 and 248. Because the exponent of 2 is 3 and the exponent of 31 is 1, the number of factors is (3 + 1)(1 + 1) = 8, which matches the list.

所以正因数完整列表为1、2、4、8、31、62、124和248。因为2的指数是3，31的指数是1，因数个数为(3+1)(1+1)=8，与列表一致。

Multiples of 248 are found by multiplying 248 by positive integers: 248, 496, 744, 992, 1240, … In IGCSE questions, you may need to find the lowest common multiple or decide whether a number can be divided equally; factor knowledge is essential.

248的倍数用正整数乘248得到：248、496、744、992、1240……在IGCSE考试中，可能需要求最小公倍数或判断一个数能否被均分；因数知识至关重要。

4. HCF and LCM | 最大公因数与最小公倍数

The highest common factor (HCF) of two numbers is the largest integer that divides both. The lowest common multiple (LCM) is the smallest positive integer that is a multiple of both. Compare 248 with 124:

两个数的最大公因数(HCF)是能同时整除它们的最大整数；最小公倍数(LCM)是同时为它们倍数的最小正整数。现在我们比较248和124：

248 = 2³ × 31, 124 = 2² × 31

Using the prime factor method, the HCF is formed from the lowest power of each common prime: 2² × 31 = 124. The LCM is formed from the highest power of each prime appearing in either number: 2³ × 31 = 248.

用质因数法，取每个共同质因数的最低次幂构成HCF：2²×31=124；取每个出现在任意一个数中的质因数的最高次幂构成LCM：2³×31=248。

In this case 124 divides 248, so the HCF is the smaller number and the LCM is the larger number. This is a useful shortcut in IGCSE exams.

本例中124能整除248，所以HCF是较小的数，LCM是较大的数。这是IGCSE考试中很有用的快捷结论。

5. Powers and Roots | 幂与根

The prime factorisation of 248 shows that it is not a perfect square, because not all exponents in 2³ × 31 are even. Therefore √248 is irrational. We can simplify it:

248的质因数分解表明它不是一个完全平方数，因为2³×31中并非所有指数都是偶数。因此√248是无理数。我们可以化简：

√248 = √(4 × 62) = 2√62 ≈ 15.748

To estimate the square root without a calculator, compare with neighbouring squares: 15² = 225 and 16

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Mastering IGCSE Maths: A Study of 248 | IGCSE数学精通：数字248研究