What are all the factors, the prime factorization, and factor pairs of 45?

Factors are the numbers we multiply together to get a product. Understanding these factors sheds light on the structure and properties of a given number within mathematics. Using 45 as an illustrative example, factors of a number are those whole numbers that when multiplied by another whole number give the product as the original number, illustrating the importance of factors in understanding the divisibility and multiplicative aspects of 45.

The concept of factors for 45 relates to all the whole numbers that can divide 45 without leaving any remainder. To visually represent the factors of 45, we can list them along with their pairings that contribute to the multiplication process resulting in 45.

• 1 x 45 = 45
• 3 x 15 = 45
• 5 x 9 = 45

The above multiplication pairs showcase the factors of 45, clearly demonstrating how each factor pair multiplies to the original number 45.

To determine the factors of 45, we utilize the division method. This entails dividing 45 by integers to discover which ones result in whole numbers. Any divisor that produces a whole number quotient can be considered a factor of 45.

Let’s apply the division method:

• 45 ÷ 1 = 45, hence 1 is a factor.
• 45 ÷ 3 = 15, hence 3 is a factor.
• 45 ÷ 5 = 9, hence 5 is a factor.
• 45 ÷ 9 = 5, hence 9 is a factor.
• 45 ÷ 15 = 3, hence 15 is a factor.
• 45 ÷ 45 = 1, hence 45 is a factor.

According to the division method, any number that does not divide 45 completely, that is leaves a remainder, is not considered a factor.

Pair factors of 45 consist of two numbers that when multiplied together result in the number 45. Below is a list of both positive and negative pair factors:

This list encompasses all the possible pair factors of 45, demonstrating the concept comprehensively.

Prime factorization entails breaking down the number 45 into its prime factors. Prime factors of a number are prime numbers that multiply to result in the original number. The prime factorization of 45 can be illustrated through a factor tree, where we continue factorizing until we are only left with prime numbers:

• 45 can be divided by 5, which gives us 5 x 9
• 9 can be divided by 3, which gives us 3 x 3

So, the prime factorization of 45 is 3 x 3 x 5 or 3² x 5, where 3 and 5 are the prime factors of 45.

As we conclude our exploration of the factors of 45, here are some important points to keep in mind:

• The total number of factors for 45 is 6, which includes 1, 3, 5, 9, 15, and 45.
• The prime factors of 45 are 3 and 5.
• The positive pair factors are (1, 45), (3, 15), and (5, 9), and the corresponding negative pairs would be (-1, -45), (-3, -15), and (-5, -9).
• The prime factorization of 45 is demonstrated as 3² x 5.
• The sum of all factors of 45 is 78.

This information provides insight into the fundamental properties of the number 45 and its significance in mathematical contexts.

• Determine if 7 is a factor of 45.
• Without calculating, state if the sum of the factors of 45 is even or odd.
• Identify the prime factorization of 45 in exponential form.

1. 7 is not a factor of 45 as 45 cannot be divided equally by 7.
2. The sum of the factors of 45 is odd, which is 78.
3. The prime factorization of 45 in exponential form is 3² x 5.