Cộng tính: chia hết, dư số và biến đổi nhân
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| remainder/rɪˈmeɪndə/ | dư |
| GCF/ˌdʒiː siː ˈef/ | The greatest common factor: the largest whole number dividing each number. |
| LCM/ˌel siː ˈem/ | The least common multiple: the smallest positive number each number divides. |
Factors do the work
- "What is the least positive integer divisible by 12, 18 and 30?" Listing multiples is slow; prime factorisation is fast.
- GRE arithmetic tests divisibility, remainders 余数 and multiplicative change (ratios, percentages) — usually with a shortcut.
Prime factorisation: GCF 最大公因数 and LCM 最小公倍数
- $12 = 2^2 \cdot 3$, $18 = 2 \cdot 3^2$, $30 = 2 \cdot 3 \cdot 5$. LCM: highest powers: $2^2 \cdot 3^2 \cdot 5 = 180$. GCF: lowest common powers: $2 \cdot 3 = 6$.
- A number is divisible by $n$ if it contains all of $n$'s prime factors.
Divisible by 12?
Use prime factors to test divisibility.
What is the least common multiple of 8, 12 and 20?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$2^3 \cdot 3 \cdot 5 = 120$.
What is the greatest common factor of 84 and 126?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$84 = 2^2 \cdot 3 \cdot 7$, $126 = 2 \cdot 3^2 \cdot 7$; GCF $= 2 \cdot 3 \cdot 7 = 42$.
Remainders
- "When $n$ is divided by 7, the remainder is 3" means $n = 7k + 3$: 3, 10, 17, 24, …
- Remainders of sums follow the remainders: if $a$ leaves 3 and $b$ leaves 5 when divided by 7, $a + b$ leaves $8 - 7 = 1$.
$n$ leaves remainder 2 when divided by 5. What remainder does $3n$ leave? $3 \times 2 = 6$, and 6 leaves 1 when divided by 5. Check: $n = 7$, $3n = 21$, remainder 1 ✓.
$n$ leaves remainder 4 when divided by 6. What remainder does $2n$ leave when divided by 6?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$2 \times 4 = 8$, which leaves 2. Check $n = 10$: $20 = 18 + 2$.
Ratios and percentages as multipliers
- A 25% increase then a 20% decrease: $1.25 \times 0.8 = 1$ (no change).
- Ratio 3 : 5 of boys to girls means boys are $\tfrac{3}{8}$ of the class.
The ratio of boys to girls is 3 : 5. What fraction of the class are girls?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Girls are 5 of every 8 students.
Common traps
- Multiplying the numbers to get an LCM (12 × 18 × 30) instead of using highest prime powers.
- Treating a part-to-part ratio as part-to-whole: 3 : 5 is not $\tfrac{3}{5}$ of the class.
The LCM of 12 and 18 is 36.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. 12 = 2²·3 and 18 = 2·3², so the LCM is 2²·3² = 36. The product 216 is a common multiple but not the least.
Recap
- Factorise into primes: LCM = highest powers, GCF = lowest common powers.
- Write remainders as $dk + r$; combine percentage changes by multiplying.
- GCF: The greatest common factor: the largest whole number dividing each number.
- LCM: The least common multiple: the smallest positive number each number divides.