Please explain the answer to the first question in this set. Thank you

In order for a product of two numbers to be odd, both of the numbers must be odd. For example, 3 multiplied by 9 gives you 27, but if you had 3 multiplied by an even number like 6, the product will be even. In this case, the product is 18.

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Number concepts | Lesson

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What are number concepts?

Have you noticed that the sum of two even integers is always even? Or how only integers ending with a 5 or a 0 are divisible by 5?

Number concepts are the interesting properties that exist between numbers. These ideas help us perform calculations and solve problems.

What skills are tested?

Using the properties of odd and even numbers
Finding factors and greatest common factors
Finding multiples and least common multiples
Using number properties to solve problems

What's always true when adding or multiplying odd and even numbers?

The following properties are true whenever odd or even numbers are added or multiplied.

Addition

The sum of two odd numbers is even.
The sum of two even numbers is even.
The sum of an odd and an even number is odd.

Multiplication

The product of two odd numbers is odd.
The product of two even numbers is even.
The product of an odd and an even number is even.

Knowing about these properties is useful, but it isn't necessary to memorize them! To recall any of the above properties, create a simple example for the property you need.

What are factors and how do we find them?

Factors are whole numbers that divide evenly (no remainder) into another whole number.

To find the factors of a number, we can use a calculator or the divisibility rules.

Below are the most common divisibility rules.

Divisible by	If
$2$ ‍	Last digit is $0, 2, 4, 6,$ ‍ or $8$ ‍
$3$ ‍	Sum of digits is divisible by $3$ ‍
$4$ ‍	Last $2$ ‍ digits divisible by $4$ ‍
$5$ ‍	Last digit is $0$ ‍ or $5$ ‍
$6$ ‍	Number is divisible by $2$ ‍ and $3$ ‍
$8$ ‍	Last $3$ ‍ digits divisible by $8$ ‍
$9$ ‍	Sum of digits is divisible by $9$ ‍
$10$ ‍	Last digit is $0$ ‍

Factoring a number means listing all its factors. The factors are usually presented in a list ordered from least to greatest.

One way to create a list of factors is to start checking for factor pairs beginning at

1

and checking each subsequent integer. We can stop checking for factors when we get to an integer that's already been paired with a previous factor.

What is prime factorization?

Prime numbers are whole numbers greater than

1

whose only factors are

1

and itself. For example,

17

is prime because it is only divisible by

1

and

17

Prime factorization means writing a number as a product of factors that are all prime numbers. Prime factorization is helpful when finding the greatest common factor or least common multiple, but is not the only way to do it.

We can write

30

as the product of prime factors by finding any factor pair, then rewriting any non-prime factor as the product of prime numbers.

For example, we can write

30

as the product of its factors

2

and

15

. The number

2

is prime but

15

is not. Then, we rewrite

15

as the product of its factors

3

and

5

. We can stop here because

2

3

, and

5

are all prime.

A factor tree is a handy way to organize these steps:

The prime factorization of

30

2 \times 3 \times 5

What is the greatest common factor?

The greatest common factor (or greatest common divisor) of two numbers is the largest whole number that both numbers are divisible by.

We can find the greatest common factor one of two ways:

List the factors of each number and select the largest common factor.
Write the prime factorization of each number and then multiply all common factors.

A coach wants to divide

24

pitchers,

36

outfielders, and

60

infielders into teams such that each team has the same number of each type of player. What is the greatest number of teams that can be formed?

We can list the factors of each number:

$24$ ‍: $1$ ‍, $2$ ‍, $3$ ‍, $4$ ‍, $6$ ‍, $8$ ‍, $12$ ‍, $24$ ‍

$36$ ‍: $1$ ‍, $2$ ‍, $3$ ‍, $4$ ‍, $6$ ‍, $9$ ‍, $12$ ‍, $18$ ‍, $36$ ‍
$60$ ‍: $1$ ‍, $2$ ‍, $3$ ‍, $4$ ‍, $5$ ‍, $6$ ‍, $10$ ‍, $12$ ‍, $15$ ‍, $20$ ‍, $30$ ‍, $60$ ‍

The greatest common factor is

12

Another approach is to write the prime factorization of each number. The greatest common factor is the product of the common factors:

$24 = 2 \times 2 \times 2 \times 3$ ‍
$36 = 2 \times 2 \times 3 \times 3$ ‍
$60 = 2 \times 2 \times 3 \times 5$ ‍

The greatest common factor is

2 \times 2 \times 3 = 12

This means the coach can form at most

12

teams. (Each team will have

24 \div 12 = 2

pitchers,

36 \div 12 = 3

outfielders, and

60 \div 12 = 5

infielders.)

What are multiples?

A multiple is a number that results when we multiply a whole number by another non-zero whole number.

To find the first few multiples of a number, multiply the number by whole numbers starting with

1

What is the least common multiple?

The least common multiple of two numbers is the smallest whole number that is divisible by both numbers.

We can find the least common multiple one of two ways:

List the multiples of each number until we find a common multiple.
Multiply together all the unique factors in the prime factorization of both numbers.

Hots dogs come in packages of

10

and hot dog buns come in packages of

8

. What is the least number of packages of hot dogs Caleb can buy in order to get the same number of hot dogs and hot dog buns?

We can list the first few multiples of each number:

$8$ ‍: $8$ ‍, $16$ ‍, $24$ ‍, $32$ ‍, $40$ ‍, $48$ ‍, $56$ ‍

$10$ ‍: $10$ ‍, $20$ ‍, $30$ ‍, $40$ ‍, $50$ ‍, $60$ ‍

The least common multiple is

40

Another approach is to write the prime factorization of each number. The least common multiple is the product of the unique factors of both numbers. For repeated factors, the factor is used as many times as it appears in the factorization in which it appears the most times.

$8 = 2 \times 2 \times 2$ ‍
$10 = 2 \times 5$ ‍

For example,

2

appears three times in the prime factorization of

8

and

5

appears once in the prime factorization of

10

. Therefore, the least common multiple of

8

and

10

2 \times 2 \times 2 \times 5 = 40

This means that Caleb needs to buy

40

hot dogs and

40

buns. To get

40

hot dogs, Caleb needs to buy

40 \div 10 = 4

packages of hot dogs.

The least number of packages of hot dogs Caleb can buy is

4

Your turn!

TRY: USING NUMBER PROPERTIES

The product of an odd and an even number is even.

The product of two even numbers is even.

The product of $k$ ‍ and $r$ ‍ is odd.

According to the statements above, which of the following is a valid conclusion?

TRY: FINDING COMMON FACTORS

Which of the following are factors of both

15

and

45

TRY: APPLYING NUMBER PROPERTIES

Jasmine is working on an art project. She has one piece of construction paper that is

21 cm

wide and a second piece that is

33 cm

wide. Jasmine wants to cut both pieces of paper into strips that are equal in width and as wide as possible. How wide should Jasmine cut each strip?

cm

TRY: APPLYING NUMBER PROPERTIES

Two of the lights in Sarah's homeroom are flickering. They both just flickered at the same time. One of the lights flickers every

7

seconds and the other light flickers every

8

seconds. How many seconds will pass before both lights flicker at the same time again?

seconds

Things to remember

We can determine whether the sum or product of two integers is even or odd using the properties of even and odd numbers.

The greatest common factor of two numbers is the largest whole number that both numbers are divisible by.

The least common multiple of two numbers is the smallest whole number that is divisible by both numbers.

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Debbie Smothers
Posted 4 years ago. Direct link to Debbie Smothers's post “Please explain the answer...”
Please explain the answer to the first question in this set.
Thank you
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- Steven Parkmond II
  Posted 3 years ago. Direct link to Steven Parkmond II's post “In order for a product of...”
  In order for a product of two numbers to be odd, both of the numbers must be odd. For example, 3 multiplied by 9 gives you 27, but if you had 3 multiplied by an even number like 6, the product will be even. In this case, the product is 18.
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andy
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monke
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Annie Williams
Posted 3 years ago. Direct link to Annie Williams's post “how do you get to factor?...”
how do you get to factor? thank you
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shatha
Posted a year ago. Direct link to shatha's post “why does it take for ever...”
why does it take for ever to scroll while im using the computer
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Praxis Core Math

Course: Praxis Core Math > Unit 1