Chapter 4.2, Problem 11E

1.

To determine

To find the greatest number of identical bags an owner can make out of crayons, paintbrushes and oil paints.

Answer to Problem 11E

Prime Factors of 225 are:3, 3, 5, 5.

Prime Factors of 75 are:3, 5, 5.

Prime Factors of 120 are:2,2,2, 3, 5.

Explanation of Solution

Given Information:

225 crayons, 75 paintbrushes and 120 tubes of oil paint.

Formula used: using greatest common factor

Calculation:

1. To get the Prime Factors of 225, divide 225 by the smallest prime number possible. Then take the result from that and divide that by the smallest prime number. Repeat this process until end up with 1.

This Prime Factorization process creates the Prime Factor Tree of 225. All the prime numbers that are used to divide in the Prime Factor Tree are the Prime Factors of 225. Here is the math to illustrate:

  $\begin{array}{l}225÷3=75\\ 75÷3=25\\ 25÷5=5\\ 5÷5=1\end{array}$

Again, all the prime numbers used to divide above are the Prime Factors of 225.

• Thus, the Prime Factors of 225 are:

3, 3, 5, 5.

Repeat the process for the other two numbers.

• The Prime Factors of 75 are:

3, 5, 5.

• The Prime Factors of 120 are:

2, 2, 2, 3, 5.

2.

To determine

To find GCF & Common Prime Factors

Answer to Problem 11E

The common factors are: 1, 3, 5, 15

Then the greatest common factor is 15.

Explanation of Solution

Given Information:

225 crayons, 75 paintbrushes and 120 tubes of oil paint.

Formula used: using greatest common factor

Calculation:

The factors of 75 are: 1, 3, 5, 15, 25, 75

The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

The factors of 225 are: 1, 3, 5, 9, 15, 25, 45, 75, 225

The common factors are: 1, 3, 5, 15

Then the greatest common factor is 15.

3.

To determine

To understand what the GCF represents

Answer to Problem 11E

The greatest number of identical gift-bags that can be made out of 225 crayons, 75 paintbrushes and 120 tubes of oil paint

Explanation of Solution

Given Information:

225 crayons, 75 paintbrushes and 120 tubes of oil paint.

Formula used: using greatest common factor

Calculation:

The greatest common factor here represents the greatest number of identical gift-bags that can be made out of 225 crayons, 75 paintbrushes and 120 tubes of oil paint.