Answer
a) $40 = 2\cdot 2 \cdot 2 \cdot 5$
b) $40 = 2\cdot 2 \cdot 2 \cdot 5$
c) Similarities: same prime factorization. Differences: the order in which the prime factors are determined.
Work Step by Step
a) To write the prime factorization of $40$ using $2$ and $20$ as the first pair of factors, we can start by expressing $20$ as a product of its prime factors:
$20 = 2 \cdot 2 \cdot 5$
Then, we can write $40$ as the product of $2$ (from the first pair of factors) and the prime factorization of $20$:
$40 = 2\cdot (2\cdot 2 \cdot 5)$
Simplifying further, we get:
$40 = 2\cdot 2\cdot 2\cdot 5$
b) To write the prime factorization of $40$ using $4$ and $10$ as the first pair of factors, we need to express $4$ and $10$ as products of their prime factors:
$4 = 2 \cdot 2$
$10 = 2 \cdot 5$
Then, we can write $40$ as the product of $4$ (from the first pair of factors) and the prime factorization of $10$:
$40 = (2 \cdot 2) \cdot (2 \cdot 5)$
Simplifying further, we get:
$40 = 2\cdot 2\cdot 2 \cdot 5$
c) In both parts a) and b), the prime factorization of $40$ is expressed as $2\cdot 2\cdot 2\cdot 5$.
The difference lies in the arrangement of the factors.
In part a), the first pair of factors used is $2$ and $20$. The factorization starts with $20$, which is then broken down into its prime factors ($2 \cdot 2 \cdot 5$). This is followed by multiplying $2$ (from the first pair) with the prime factorization of $20$.
In part b), the first pair of factors used is $4$ and $10$. The factorization starts with $4$ (already expressed as a product of its prime factors, $2 \cdot 2$), and then the prime factorization of $10$ ($2\cdot 5$) is multiplied with it. The differences lie in the arrangement and the way the factorization is presented, depending on the initial pair of factors used.
