What are all the factors, the prime factorization, and factor pairs of 70?

Factors are the fundamental numbers we multiply together to obtain a product, representing a fascinating aspect of a number’s structure within mathematics. Taking 70 as an illustrative example, factors are those numbers that, when multiplied in various combinations, yield the product 70. Understanding these factors sheds light on the inherent properties and characteristics of 70 and its role in various mathematical scenarios.

Factors of 70 are the whole numbers that can be coupled to form a product equal to 70. These pairs of factors show the different combinations that when multiplied, represent the number 70.

Here is a list of factors for 70:

• 1
• 2
• 5
• 7
• 10
• 14
• 35
• 70

To find the factors of 70, we can use the division method similar to how we determined the factors of 24 in the examples provided:

• Start with 1, which is a factor of all natural numbers, including 70. Thus, 70 ÷ 1 = 70.
• Proceed to divide 70 by integers increasing from 1 up to 70 itself. If the result of the division is a whole number, then both the divisor and the quotient are factors of 70.
• Since 70 is an even number, it is divisible by 2. So, 70 ÷ 2 = 35, making 2 a factor of 70.
• Continue this process with other numbers such as 5 and 7, both of which are factors of 70, as 70 ÷ 5 = 14, and 70 ÷ 7 = 10.

Using this division process, you will find that the numbers 1, 2, 5, 7, 10, 14, 35, and 70 are all factors of 70 since dividing 70 by each of these numbers returns a whole number as the quotient.

Pair factors of 70 are two numbers which when multiplied together yield 70. These can be presented as positive and negative pair factors.

Here is a table of positive pair factors:

• 1 × 70 = 70
• 2 × 35 = 70
• 5 × 14 = 70
• 7 × 10 = 70

And similarly, we can derive the negative pair factors:

• -1 × -70 = 70
• -2 × -35 = 70
• -5 × -14 = 70
• -7 × -10 = 70

The prime factorization of 70 entails breaking it down to its basic prime factors. This process helps to uncover the number’s simplest building blocks. A factor tree diagram can represent these steps visually.

The prime factors of 70 are determined as follows:

• Divide 70 by the smallest prime number possible, which is 2. We then get 70 ÷ 2 = 35.
• 35 is not divisible by 2, so we proceed to the next smallest prime number, which is 5. We find that 35 ÷ 5 = 7.
• 7 is a prime number, and so our process is complete, resulting in 70 = 2 × 5 × 7.

Thus, the prime factorization of 70 is represented as 2 × 5 × 7.

Let’s summarize some important points to keep in mind when it comes to factors of 70:

• The total count of factors for 70 is 8.
• The prime factors of 70 are 2, 5, and 7.
• The sum of all factors of 70, including both positive and negative, is 140.
• 70 has four pair factors: (1, 70), (2, 35), (5, 14), and (7, 10).

Engage with the following exercises to deepen your understanding of the factors of 70:

• Exercise 1: Calculate the sum of all the positive factors of 70.
• Exercise 2: Identify a pair of factors of 70 that are both odd numbers.
• Exercise 3: List all the prime factors of 70 without repeating any numbers.

Here are the solutions to the exercises provided in the previous section:

• Solution to Exercise 1: The sum of the positive factors of 70 is 1+2+5+7+10+14+35+70 = 144.
• Solution to Exercise 2: The pair of factors of 70 that are both odd numbers is (5, 14).
• Solution to Exercise 3: The list of all prime factors of 70 without repetition is 2, 5, 7.