Find the Least Common Multiple (LCM) of the given algebraic expressions and numbers.

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4)

Factorize each expression.
$x^2 + x = x(x+1)$ $x^2 - 1 = (x-1)(x+1)$ $x^2 - x = x(x-1)$
Identify all unique factors and their highest powers.
The unique factors are $x$ , $(x+1)$ , and $(x-1)$ . Each appears with a highest power of 1.
Multiply these highest powers together to find the L.C.M.
$\text{L.C.M} = x(x+1)(x-1)$
This can also be written as:
$\text{L.C.M} = x(x^2-1) = x^3-x$ $\boxed{x(x^2-1) \text{ or } x^3-x}$
5)
Find the prime factorization of each number.
$9 = 3^2$ $15 = 3 \times 5$ $20 = 2^2 \times 5$ $24 = 2^3 \times 3$
Identify all unique prime factors and their highest powers.
- Highest power of 2: $2^3$
- Highest power of 3: $3^2$
- Highest power of 5: $5^1$
Multiply these highest powers together to find the L.C.M.
$\text{L.C.M} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 72 \times 5 = 360$
The least number of sweets is 360.
$\boxed{360}$
6)
Find the prime factorization of each number.
$27 = 3^3$ $30 = 2 \times 3 \times 5$ $45 = 3^2 \times 5$
Identify all unique prime factors and their highest powers.
- Highest power of 2: $2^1$
- Highest power of 3: $3^3$
- Highest power of 5: $5^1$
Multiply these highest powers together to find the L.C.M.
$\text{L.C.M} = 2^1 \times 3^3 \times 5^1 = 2 \times 27 \times 5 = 54 \times 5 = 270$
Add the remainder to the L.C.M.
to find $n$ .
Since the remainder is always 3, $n = \text{L.C.M} + 3$ .
$n = 270 + 3 = 273$ $\boxed{273}$
7)
Find the prime factorization of each number (20, 24, 28).
$20 = 2^2 \times 5$ $24 = 2^3 \times 3$ $28 = 2^2 \times 7$
Identify all unique prime factors and their highest powers.
- Highest power of 2: $2^3$
- Highest power of 3: $3^1$
- Highest power of 5:

Find the Least Common Multiple (LCM) of the given algebraic expressions and numbers.

Step-by-step solution

Factorize each expression.

Identify all unique factors and their highest powers.

Multiply these highest powers together to find the L.C.M.

Find the prime factorization of each number.

Identify all unique prime factors and their highest powers.

Multiply these highest powers together to find the L.C.M.

Find the prime factorization of each number.

Identify all unique prime factors and their highest powers.

Multiply these highest powers together to find the L.C.M.

Add the remainder to the L.C.M.

Find the prime factorization of each number (20, 24, 28).

Identify all unique prime factors and their highest powers.

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