What are all the factors, the prime factorization, and factor pairs of 42?

Factors are integral to understanding the composition of numbers within the realm of mathematics. They are the building blocks that, when multiplied together, produce a given number. For our illustrative example, the number in focus is 42. Comprehending the factors of 42 provides insight into its structure and the various ways it can be integrated and utilized in mathematical expressions and real-world applications.

Factors of 42 are the integers which, when multiplied in pairs, give the product 42. This means that each factor pair combines to make the number 42. Below is a representation of these factors and their pairings:

• 1 × 42 = 42
• 2 × 21 = 42
• 3 × 14 = 42
• 6 × 7 = 42

The comprehensive list of factors for 42 is depicted in the following table:

The process of discovering the factors of 42 via division is straightforward. Dividing 42 by an integer that results in a whole number indicates that the divisor is a factor of 42. Here’s an example to illustrate the methodology:

• 42 ÷ 1 yields 42, thus 1 is a factor.
• 42 ÷ 2 yields 21, thus 2 is a factor.
• 42 ÷ 3 yields 14, thus 3 is a factor.
• The division process continues with integers up to 42.

Factors of 42 must be whole numbers and cannot be fractions or decimals. If the division does not result in a whole number, it’s not a factor of 42. For instance:

42 ÷ 5 = 8.4, which means 5 is not a factor.

Thus, we conclude that the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

Pair factors are two numbers that, when multiplied together, equal 42. These can be listed as both positive and negative pairs because a negative multiplied by a negative yields a positive. Here are the positive and negative pair factors of 42 presented in tabular form:

The prime factorization of 42 involves breaking it down into its basic prime elements. A factor tree can help visualize the step-by-step decomposition:

1. Start by dividing 42 by the smallest prime number possible, which is 2, yielding 21.
2. Since 21 is not divisible by 2, we move to the next smallest prime, which is 3, yielding 7.
3. 7 is a prime number and cannot be divided further.

The prime factorization of 42 is then 2 × 3 × 7.

Note: Factor tree diagrams can vary as they might begin with different factor pairs, but the end prime factors will always be the same.

There are several key aspects to remember when considering the factors of 42:

• 42 has a total of 8 factors: 1, 2, 3, 6, 7, 14, 21, and 42.
• The prime factors of 42 are 2, 3, and 7.
• The pair factors of 42 include both positive pairs (e.g. 6 and 7) and corresponding negative pairs (e.g. -6 and -7).
• The sum of all the factors of 42 is 96 (1+2+3+6+7+14+21+42).

These points provide a comprehensive summary of the unique qualities and properties of the number 42, most notably its factors.