1.质因数分解题：
Express
396
396
as a product of prime factors using index notation.

质因数分解题

  1. Express
    432
    432
    as a product of prime factors using index notation.
  2. Given
    N = 2^3 \times 3^2 \times 5
    N = 2^3 \times 3^2 \times 5
    , calculate the number of positive divisors of
    N
    N
    .
  3. Two numbers have prime factorizations:
    18 = 2 \times 3^2
    18 = 2 \times 3^2
    and
    24 = 2^3 \times 3
    24 = 2^3 \times 3
    . Find their GCD (greatest common divisor) and LCM (least common multiple).

同余问题

  1. Find the smallest positive integer that leaves a remainder of
    3
    3
    when divided by
    4
    4
    and by
    7
    7
    .
  2. A number leaves remainder
    5
    5
    when divided by
    8
    8
    and by
    12
    12
    . What is the smallest such number?
  3. Find the largest three - digit number that leaves remainder
    2
    2
    when divided by
    5
    5
    and by
    9
    9
    .
  4. A number gives remainder
    4
    4
    when divided by
    6
    6
    , and remainder
    7
    7
    when divided by
    9
    9
    (hint: both remainders are
    2
    2
    less than the divisors). Find its smallest positive value.

整除性问题

  1. Check if
    738
    738
    is divisible by
    9
    9
    (use the divisibility rule for
    9
    9
    : sum of digits).
  2. Find the smallest four - digit number divisible by
    15
    15
    .
  3. Using digits
    2
    2
    ,
    5
    5
    ,
    7
    7
    , form a three - digit number divisible by both
    2
    2
    and
    5
    5
    .
  4. How many two - digit numbers are divisible by
    3
    3
    but not by
    6
    6
    ?

综合应用（质因数 + 同余/整除）

  1. The GCD of two numbers is
    12
    12
    (
    2^2 \times 3
    2^2 \times 3
    ) and their LCM is
    72
    72
    (
    2^3 \times 3^2
    2^3 \times 3^2
    ). If one number is
    24
    24
    (
    2^3 \times 3
    2^3 \times 3
    ), find the other number.
  2. A number
    n
    n
    has: remainder
    1
    1
    when divided by
    4
    4
    , and remainder
    3
    3
    when divided by
    6
    6
    . Show
    n + 3
    n + 3
    is divisible by both
    4
    4
    and
    6
    6
    , then find the smallest positive
    n
    n
    .

2. 坐标变换：

原题
A(1, -4)
A(1, -4)
is rotated anticlockwise about the origin through
90^\circ
90^\circ
to
A'
A
. Find the coordinates of
A'
A
.

旋转类

  1. B(3, 2)
    B(3, 2)
    is rotated anticlockwise about the origin through
    90^\circ
    90^\circ
    to
    B'
    B
    . Find the coordinates of
    B'
    B
    .
  2. C(-2, 5)
    C(-2, 5)
    is rotated clockwise about the origin through
    90^\circ
    90^\circ
    to
    C'
    C
    . Find the coordinates of
    C'
    C
    .
  3. D(4, -1)
    D(4, -1)
    is rotated about the origin through
    180^\circ
    180^\circ
    to
    D'
    D
    . Find the coordinates of
    D'
    D
    .

坐标平移题

  1. Point
    P(5, -3)
    P(5, -3)
    is translated by the vector
    \begin{pmatrix} 2 \\ 4 \end{pmatrix}
    \begin{pmatrix} 2 \\ 4 \end{pmatrix}
    . Find the coordinates of the image
    P'
    P
    .
  2. Point
    Q(-1, 6)
    Q(-1, 6)
    is translated 3 units to the left and 5 units down. Find the coordinates of
    Q'
    Q
    .
  3. The image of point
    R(a, b)
    R(a, b)
    after translation is
    R'(a+4, b-2)
    R
    . Describe the translation in words.

坐标翻折（轴对称）题

  1. Find the image of
    A(2, -5)
    A(2, -5)
    when reflected in the x-axis.
  2. Point
    B(-3, 4)
    B(-3, 4)
    is reflected in the y-axis. Find its image
    B'
    B
    .
  3. Find the coordinates of the image of
    C(1, 7)
    C(1, 7)
    after reflection in the line
    y = x
    y = x
    .
  4. Reflect point
    D(-2, 3)
    D(-2, 3)
    in the line
    y = -x
    y = -x
    . Find the image
    D'
    D
    .
  5. Reflect point
    E(5, 1)
    E(5, 1)
    in the line
    x = 2
    x = 2
    . Find the coordinates of
    E'
    E
    .
  6. Point
    F(0, -1)
    F(0, -1)
    is reflected in the line
    y = 3
    y = 3
    . Find its image
    F'
    F
    .

第10题（利润计算）：
The cost of the cup of coffee is

$15
$15
. If it is now sold at a profit of
120\%
120\%
, find the selling price of the cup of coffee.

第11题（折扣计算）：
The marked price of a box of face masks is

\$125
\$125
aand it is sold at a discount of
30\%
30\%
on the marked price. Find the discount.

1. 折扣相关（Discount）

  • 标价（Marked Price, MP）：商品标注的原价（比如“标价$125”）。
  • 折扣率（Discount Rate）：按标价的百分比降价（比如“30%折扣”，即降标价的30%）。
  • 折扣金额（Discount）：实际少收的钱，公式：
    \text{Discount} = \text{MP} \times \text{Discount Rate}
    \text{Discount} = \text{MP} \times \text{Discount Rate}
  • 售价（Selling Price, SP，折扣后）：标价减去折扣，公式：
    \text{SP} = \text{MP} - \text{Discount} = \text{MP} \times \left(1 - \frac{\text{Discount Rate}}{100}\right)
    \text{SP} = \text{MP} - \text{Discount} = \text{MP} \times \left(1 - \frac{\text{Discount Rate}}{100}\right)

2. 利润相关（Profit）

  • 成本价（Cost Price, CP）：商家进货的成本（比如“成本$15”）。
  • 利润（Profit）：售价减去成本（赚的钱，若售价<成本则为亏损Loss），公式：
    \text{Profit} = \text{SP} - \text{CP} \quad (\text{若SP>CP，否则Loss=CP−SP})
    \text{Profit} = \text{SP} - \text{CP} \quad (\text{若SP>CP，否则Loss=CP−SP})
  • 利润率（Profit Rate）：利润占成本的百分比（亏损率同理，分母是CP），公式：
    \text{Profit Rate} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\%
    \text{Profit Rate} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\%
  • 已知利润率求售价：若利润率为
    r\%
    r\%
    （基于成本），则售价 = 成本 + 利润，即：
    \text{SP} = \text{CP} \times \left(1 + \frac{r}{100}\right)
    \text{SP} = \text{CP} \times \left(1 + \frac{r}{100}\right)

因式分解

第29题：
(a) Factorize

x^2 + 22x + 121
x^2 + 22x + 121
.
(b) Hence, factorize
(a - b)^2 + 22a - 22b + 121
(a - b)^2 + 22a - 22b + 121
.

第30题：
(a) Factorize

(x + y)^4 - (x - 3y)^4
(x + y)^4 - (x - 3y)^4
.
(b) Hence, factorize
(4a^2 + 9b^2)^4 - (4a^2 - 27b^2)^4
(4a^2 + 9b^2)^4 - (4a^2 - 27b^2)^4

同类题

一、完全平方公式

(a) 基础完全平方因式分解

  1. Factorize
    x^2 - 18x + 81
    x^2 - 18x + 81
    .
  2. Factorize
    4x^2 + 20x + 25
    4x^2 + 20x + 25
    .
  3. Factorize
    9y^2 - 42y + 49
    9y^2 - 42y + 49
    .

(b) 换元

  1. Hence, factorize
    (m + n)^2 - 18(m + n) + 81
    (m + n)^2 - 18(m + n) + 81
    .
  2. Hence, factorize
    (2a - b)^2 + 20(2a - b) + 25
    (2a - b)^2 + 20(2a - b) + 25
    .
  3. Hence, factorize
    (3x + 2y)^2 - 42(3x + 2y) + 49
    (3x + 2y)^2 - 42(3x + 2y) + 49
    .

二、四次方差分解 + 换元法

利用平方差公式：
A^4 - B^4 = (A^2 + B^2)(A + B)(A - B)
A^4 - B^4 = (A^2 + B^2)(A + B)(A - B)
）

(a) 基础四次方差因式分解

  1. Factorize
    (2x + y)^4 - (x - 2y)^4
    (2x + y)^4 - (x - 2y)^4
    .
  2. Factorize
    (3a - b)^4 - (a + 3b)^4
    (3a - b)^4 - (a + 3b)^4
    .
  3. Factorize
    (5m + 2n)^4 - (2m - 5n)^4
    (5m + 2n)^4 - (2m - 5n)^4
    .

(b) 换元应用（代入复杂二项式，复用(a)的分解逻辑）

  1. Hence, factorize
    (x^2 + 2y^2)^4 - (x^2 - 3y^2)^4
    (x^2 + 2y^2)^4 - (x^2 - 3y^2)^4
    .
  2. Hence, factorize
    (2p^2 + 3q^2)^4 - (2p^2 - 9q^2)^4
    (2p^2 + 3q^2)^4 - (2p^2 - 9q^2)^4
    .
  3. Hence, factorize
    (5a^3 + b^3)^4 - (5a^3 - 2b^3)^4
    (5a^3 + b^3)^4 - (5a^3 - 2b^3)^4
    .


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