Question

1. for each of the following sums, identify the greatest common factor of both terms and use the distributive property to write the sum as a product. evaluate the product and check your answer with the original sum.

(a) 21 + 49
(b) 45 + 27

using your math

1. a rectangle with a total of 80 square feet has been broken into two rectangles with areas of 56 square feet and 24 square feet as shown. the common length of the two smaller rectangles is a whole number that is greater than either width 1 or width 2, which are both whole numbers. from this information, determine the length, width 1, and width 2. explain how you found your answers.

(diagram: two rectangles side by side, labeled length, width 1 (56 sq. ft), width 2 (24 sq. ft))

reviewing your math

1. find the least common multiple of each of the following pairs of numbers.

(a) 4 and 10
(b) 3 and 12
(c) 4 and 9
(d) 8 and 12

1. list all whole number factors of 24.

Explanation:

Problem 5(a): \(21 + 49\)

Step 1: Find GCF of 21 and 49

Factors of 21: \(1, 3, 7, 21\)
Factors of 49: \(1, 7, 49\)
GCF is \(7\).

Step 2: Apply Distributive Property

\(21 + 49 = 7 \times 3 + 7 \times 7 = 7(3 + 7)\)

Step 3: Evaluate Product

\(7(3 + 7) = 7 \times 10 = 70\)
Original sum: \(21 + 49 = 70\) (matches).

Problem 5(b): \(45 + 27\)

Step 1: Find GCF of 45 and 27

Factors of 45: \(1, 3, 5, 9, 15, 45\)
Factors of 27: \(1, 3, 9, 27\)
GCF is \(9\).

Step 2: Apply Distributive Property

\(45 + 27 = 9 \times 5 + 9 \times 3 = 9(5 + 3)\)

Step 3: Evaluate Product

\(9(5 + 3) = 9 \times 8 = 72\)
Original sum: \(45 + 27 = 72\) (matches).

Problem 6: Rectangle Dimensions

Step 1: Find GCF of 56 and 24

Factors of 56: \(1, 2, 4, 7, 8, 14, 28, 56\)
Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)
GCF is \(8\) (length, as it’s > widths).

Step 2: Find Widths

Width 1: \(\frac{56}{8} = 7\)
Width 2: \(\frac{24}{8} = 3\)

Problem 7(a): LCM of 4 and 10

Answer:

s:
5(a) GCF=7, Product= \(7(3 + 7) = 70\)
5(b) GCF=9, Product= \(9(5 + 3) = 72\)
6 Length=8, Width 1=7, Width 2=3
7(a) 20; (b) 12; (c) 36; (d) 24
8 Factors: \(1, 2, 3, 4, 6, 8, 12, 24\)