What are all the factors, the prime factorization, and factor pairs of 35?

Factors are the building blocks that multiply together to form a product. With 35 as an illustrative example, understanding the factors of a number like 35 gives insight into its mathematical composition and relationships. In essence, the factors of a number are the integers which, when multiplied together, produce the number itself.

The factors of 35 are the numbers that can be evenly divided into 35. Factors come in pairs since when two numbers multiply together, they produce the product.

Thus, the factors of 35 are 1, 5, 7, and 35.

The factors of any number are determined through a process known as division. To find the factors of 35:

• Begin by dividing 35 by 1, which is a factor for all whole numbers.
• Proceed by testing divisibility with other integers less than or equal to 35.
• If the division results in a whole number with no remainder, record both the divisor and the quotient as factors.

Here’s how you calculate:

35 ÷ 1 = 35
35 ÷ 5 = 7
35 ÷ 7 = 5
35 ÷ 35 = 1

Other integer divisors will result in fractions which do not count as factors. For instance:
35 ÷ 3 = 11.666… (not a factor)

Exploring the concept of pair factors furthers our understanding of the number 35’s divisibility. Pair factors are sets of two numbers that, when multiplied, equate to the number in question.

The positive pair factors of 35 are:

• (1, 35)
• (5, 7)

Negative pair factors also exist since the product of two negative numbers is positive:

• (-1, -35)
• (-5, -7)

The prime factorization of 35 involves breaking down the number into its basic prime factors. The prime numbers that multiply to make 35 are simply 5 and 7. A factor tree diagram can illustrate this process:

35

⬍ ⬎
5   7

Both 5 and 7 are prime numbers and thus represent the prime factors of 35.

When discussing the factors of 35, it’s vital to keep in mind the following:

• 35 has a total of four factors: 1, 5, 7, and 35.
• The prime factors of 35 are 5 and 7 only.
• The pair factors are (1, 35) and (5, 7), along with their negative counterparts.
• The sum of all the factors of 35 is 48 (1+5+7+35).

Through the exploration of factors, 35’s unique properties as a product of two distinct prime numbers become apparent.

• 1. Identify all the factors of 35.
• 2. What prime numbers are factors of 35?
• 3. Can the number 14 be a factor of 35?

1. The factors of 35 are 1, 5, 7, and 35.

2. The prime factors of 35 are 5 and 7.

3. No, 14 cannot be a factor of 35 as 35 is not divisible by 14 without leaving a remainder.