Question

factors and multiples: section a checkpoint

1. what are the possible side lengths of a rectangle with area 10 square units?

draw an example of each possible rectangle on the grid.
grid image

1. select all true statements.

a. 5 is a factor of 35.
b. 35 is a factor of 5.
c. 5 is a multiple of 35.
d. 35 is a multiple of 5.

1. blurred a prime number or a composite number? explain how you know.

Explanation:

Question 1

Step 1: Recall the formula for the area of a rectangle

The area of a rectangle is given by the formula $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width. We know that $A = 10$ square units. So we need to find pairs of positive integers $l$ and $w$ such that $l\times w=10$.

Step 2: Find the factor pairs of 10

We start by finding the factors of 10. The factors of 10 are the numbers that divide 10 without leaving a remainder.

• When $l = 1$, then from $l\times w=10$, we have $1\times w = 10$, so $w = 10$.
• When $l=2$, then $2\times w = 10$, and $w=\frac{10}{2}=5$.
• Since multiplication is commutative ($l\times w=w\times l$), the pairs (5, 2) and (10, 1) are also valid as length - width pairs.

Question 2

Step 1: Recall the definitions of factor and multiple

• A factor of a number $n$ is a number that divides $n$ without leaving a remainder.
• A multiple of a number $m$ is a number that can be written as $m\times k$ where $k$ is an integer.

Step 2: Analyze option A

To check if 5 is a factor of 35, we divide 35 by 5. $\frac{35}{5}=7$, which is an integer. So 5 is a factor of 35.

Step 3: Analyze option B

To check if 35 is a factor of 5, we divide 5 by 35. $\frac{5}{35}=\frac{1}{7}$, which is not an integer. So 35 is not a factor of 5.

Step 4: Analyze option C

To check if 5 is a multiple of 35, we need to see if $5 = 35\times k$ for some integer $k$. But $\frac{5}{35}=\frac{1}{7}$ is not an integer, so 5 is not a multiple of 35.

Step 5: Analyze option D (assuming the fourth option is "35 is a multiple of 5")

To check if 35 is a multiple of 5, we see if $35=5\times k$ for some integer $k$. Since $\frac{35}{5} = 7$ (an integer), 35 is a multiple of 5.

Question 3 (assuming the number is 7)

Step 1: Recall the definition of prime and composite numbers

• A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.
• A composite number is a positive integer greater than 1 that has more than two distinct positive divisors.

Step 2: Find the divisors of 7

We check the numbers that divide 7. The divisors of 7 are 1 and 7. Since 7 has exactly two distinct positive divisors (1 and itself), it is a prime number.

Answer:

The possible side - length pairs (length and width) of the rectangle are (1, 10), (2, 5), (5, 2), (10, 1) (in units).