Understanding Factors of 36

A factor of a number is a whole number that divides exactly into that number, leaving no remainder. To find the factors of 36, we look for all the whole numbers that divide 36 evenly.

We can systematically check numbers starting from 1:

• $36 \div 1 = 36$ (1 is a factor)
• $36 \div 2 = 18$ (2 is a factor)
• $36 \div 3 = 12$ (3 is a factor)
• $36 \div 4 = 9$ (4 is a factor)
• $36 \div 5$ - Does not divide evenly.
• $36 \div 6 = 6$ (6 is a factor)
• $36 \div 7$ - Does not divide evenly.
• $36 \div 8$ - Does not divide evenly.
• $36 \div 9 = 4$ (9 is a factor) - We've found 4 already, so we can stop checking once we reach a number that has already appeared as a quotient. The factors come in pairs.

The pairs of factors for 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). When listing factors, we only list each unique number once.

So, the complete set of positive factors of 36 is 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Analyzing the Given Sets of Factors

The question asks to pick out the set that forms the factors of 36 from the given options. Let's examine each option provided and check if all numbers within the set are indeed factors of 36.

Based on the analysis, Option 3, which is the set (2, 3, 4, 6, 9, 12, 18), contains numbers that are all factors of 36. While it's not the full list of factors (1 and 36 are missing), compared to the other options, this set includes a larger collection of factors from the complete set.

Conclusion on the Factors of 36 Set

The set (2, 3, 4, 6, 9, 12, 18) lists several numbers that divide 36 without a remainder. This means every number in this particular set is a factor of 36.

Revision Table: Key Math Concepts

Additional Information: Finding All Factors

To be sure you have found all positive factors of a number like 36, you can use prime factorization.

The prime factorization of 36 is $2^2 \times 3^2$.

Any factor of 36 will be in the form $2^a \times 3^b$, where 'a' can be 0, 1, or 2, and 'b' can be 0, 1, or 2. The possible values for 'a' are (0, 1, 2) - 3 options. The possible values for 'b' are (0, 1, 2) - 3 options. The total number of factors is the product of the number of options for each prime power: $3 \times 3 = 9$ factors.

Let's list them systematically using this method:

• $2^0 \times 3^0 = 1 \times 1 = 1$
• $2^1 \times 3^0 = 2 \times 1 = 2$
• $2^2 \times 3^0 = 4 \times 1 = 4$
• $2^0 \times 3^1 = 1 \times 3 = 3$
• $2^1 \times 3^1 = 2 \times 3 = 6$
• $2^2 \times 3^1 = 4 \times 3 = 12$
• $2^0 \times 3^2 = 1 \times 9 = 9$
• $2^1 \times 3^2 = 2 \times 9 = 18$
• $2^2 \times 3^2 = 4 \times 9 = 36$

The complete set of positive factors of 36 is indeed {1, 2, 3, 4, 6, 9, 12, 18, 36}. The provided option (2, 3, 4, 6, 9, 12, 18) is a subset of the complete set of factors.