What are all the factors, the prime factorization, and factor pairs of 36?

Factors are the building blocks of numbers, representing the integers that can be multiplied together to result in the original number. Take 36 as an illustrative example. Factors of 36 are numbers which, when multiplied in pairs, equal 36. Unveiling the factors of 36 not only provides insights into its mathematical essence but also has practical applications, such as in problem-solving and understanding number properties.

Factors of 36 are the integers that divide 36 without leaving a remainder. They are the whole numbers that, when paired and multiplied, reconstruct the number 36. Below is a visualization and a list of all factors of 36:

• 1 × 36 = 36
• 2 × 18 = 36
• 3 × 12 = 36
• 4 × 9 = 36
• 6 × 6 = 36

To discover the factors of 36, we utilize the division method. Here are the steps to find all the whole number factors of 36:

1. The division must result in a whole number for the divisor to be considered a factor.
2. Begin by dividing 36 by the smallest whole number, 1, and continue with integers upwards.
3. Stop once you reach the point where the original number is reached or when the quotient starts repeating.

For instance:

• 36 ÷ 1 = 36, so 1 is a factor.
• 36 ÷ 2 = 18, so 2 is a factor.
• Continue this process to find all factors.

Dividing 36 by any non-factor number will result in a non-whole number, which is not a factor. For example, dividing by 5 gives us 7.2, which is not a whole number.

The pair factors of 36 are two numbers that, when multiplied together, yield 36. They come in positive and negative pairs, as multiplication of two negatives also results in a positive number. Below is a table of these pair factors:

Prime factorization involves breaking down a number into its prime number factors. The prime factorization of 36 is illustrated through a factor tree, where 36 is divided by its smallest prime factor until only prime numbers are left. For 36, the complete factorization could be:

• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

The prime factors are the divisors that are prime numbers: 2 and 3. Therefore, the prime factorization is 2² × 3².

Key points about the factors of 36 include:

• Total factors: 9
• Prime factors: 2, 3
• Pair factors: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
• Sum of all factors: 91
• 36 is a perfect square, as it can be written as 6².

Let’s test your understanding of factors with these exercises:

1. Is 18 a factor of 36? Explain why.
2. If you split 36 into two groups with equal numbers, what are the two numbers?
3. What are some practical applications of understanding the factors of 36?

1. Yes, 18 is a factor of 36 because 36 divided by 18 equals 2, which is a whole number.
2. If you split 36 into two groups with equal numbers, each group would contain 18 units.
3. Understanding the factors of 36 can be useful in real-life situations such as organizing items, distributing goods equally among a group, and solving problems in construction and design that requires area calculations.