What are all the factors, the prime factorization, and factor pairs of 33?

Factors are the fundamental numbers we multiply together to attain a product. With 33 as our illustrative example, understanding its factors will illuminate the structure and characteristics of 33 within the realm of mathematics. Factors of a number, like 33, are integral for various calculations, including solving equations and understanding number properties.

The factors of 33 are the whole numbers which pair up through multiplication to result in 33. These factors are essential in conceptualizing how numbers break down into their constituents. Here, we present the comprehensive list of factors for the number 33.

• 1 x 33 = 33
• 3 x 11 = 33

Hence, the factors of 33 are 1, 3, 11, and 33.

To find the factors of 33, we use the division method. Factors must be whole numbers that when divided into 33, leave no remainder. The process is as follows:

• Begin by dividing 33 by the number 1, which is the smallest factor of all natural numbers, and we get 33 ÷ 1 = 33, confirming that 1 is a factor of 33.
• Next, try dividing 33 by other whole numbers in ascending order and find the divisions that result in whole numbers, as they denote factor pairs.
• Since 33 is an odd number, we do not need to divide by even numbers; instead, we test the next odd number, which is 3. We find that 33 ÷ 3 = 11, so both 3 and 11 are factors of 33.
• Continuing this method, we find no other factors since all other divisions will produce fractions.

Therefore, the complete set of factors of 33 is 1, 3, 11, and 33.

Pair factors of 33 are two numbers that, when multiplied together, equal 33. 33 has both positive and negative pair factors as detailed in the following list:

• Positive Pair: (1, 33)
• Positive Pair: (3, 11)
• Negative Pair: (-1, -33)
• Negative Pair: (-3, -11)

These pairs demonstrate the multipliers required to reach the product of 33, highlighting its divisibility by these numbers.

The prime factorization process breaks down the number 33 into its basic building blocks, which are its prime factors. Below is the step-by-step breakdown of 33 utilizing a factor tree:

• Start by dividing 33 by its lowest prime factor, which is 3, giving us 33 ÷ 3 = 11.
• Since 11 is also a prime number, we have effectively broken down 33 to its prime factors.
• The prime factors of 33 are therefore 3 and 11.

This can be expressed as the equation 33 = 3 x 11.

As we explore the factors of 33, here are some key takeaways:

• 33 has a total of four factors: 1, 3, 11, and 33.
• The prime factors of 33 are 3 and 11.
• Pair factors of 33 include both positive pairs (1, 33) and (3, 11), as well as their negative counterparts.
• The sum of all factors of 33 is 1 + 3 + 11 + 33 = 48.

These highlights provide a concise overview of the numerical properties of the number 33.

• Q1: List all positive factors of 33.
• Q2: Is the number 5 a factor of 33?
• Q3: If you have 33 apples and want to divide them equally into baskets, what are the possible numbers of baskets you could have?

• A1: The positive factors of 33 are 1, 3, 11, and 33.
• A2: No, 5 is not a factor of 33 as 33 is not divisible by 5 without leaving a remainder.
• A3: You could have either 1, 3, 11, or 33 baskets.