Question

1. find the greatest common factor of 36 and 90. show or explain which approach you used.

using your math

1. xavier has 30 apples and 54 snack bars that he is dividing up evenly in supply boxes that will contain both apples and snack bars.

(a) what is the greatest number of boxes he can use so that each box contains the same number of apples and each box contains the same number of snack bars? justify.
(b) based on your answer to (a), how many apples and how many snack bars will be in each box?

reviewing your math

1. a patio is to be shaped like the rectangle shown below whose length is 12 feet and whose width is 8 feet. how many square tiles that measure two feet by two feet will be needed to cover this patio? explain how you found your answer. (image of 2ft x 2ft tile and 12ft x 8ft rectangle)

Explanation:

Problem 4: Find the greatest common factor of 36 and 90.

Step 1: Prime Factorization of 36

Prime factorize 36: \(36 = 2\times18 = 2\times2\times9 = 2\times2\times3\times3\), so \(36 = 2^2\times3^2\).

Step 2: Prime Factorization of 90

Prime factorize 90: \(90 = 2\times45 = 2\times3\times15 = 2\times3\times3\times5\), so \(90 = 2^1\times3^2\times5^1\).

Step 3: Identify Common Prime Factors

The common prime factors of 36 and 90 are 2 and 3.

Step 4: Take the Lowest Powers of Common Factors

For 2, the lowest power is \(2^1\); for 3, the lowest power is \(3^2\).

Step 5: Calculate GCF

Multiply these together: \(2^1\times3^2 = 2\times9 = 18\).

Step 1: Understand the Problem

We need to find the greatest number of boxes such that apples and snack bars are divided evenly. This is the GCF of 30 and 54.

Step 2: Prime Factorization of 30

\(30 = 2\times15 = 2\times3\times5\), so \(30 = 2^1\times3^1\times5^1\).

Step 3: Prime Factorization of 54

\(54 = 2\times27 = 2\times3\times9 = 2\times3\times3\times3\), so \(54 = 2^1\times3^3\).

Step 4: Find GCF

Common factors: 2 (power 1) and 3 (power 1). GCF = \(2^1\times3^1 = 6\).

Step 1: Apples per Box

Total apples = 30, number of boxes = 6. So apples per box: \(30\div6 = 5\).

Step 2: Snack Bars per Box

Total snack bars = 54, number of boxes = 6. So snack bars per box: \(54\div6 = 9\).

Answer:

18

Problem 5(a): Greatest number of boxes for 30 apples and 54 snack bars