Problem 30 - Beginning and Intermediate Algebra

Question

Find all integer factors of each number. 36

Step-by-Step Solution

Answer

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Step 1 - Understand the Problem

The goal is to find all the integer factors of 36. Factors are the numbers that divide 36 evenly without leaving a remainder.

Step 2 - Start with the Smallest Factor

Begin by trying the smallest integers. Start with 1. Since 1 divides every integer, it is a factor of 36.

Step 3 - Continue Finding Factors

Proceed to the next integer, 2. Check if 36 divided by 2 leaves a remainder. Since \(36 ÷ 2 = 18\), 2 is a factor of 36.

Step 4 - Test Other Integers

Continue testing other integers in ascending order. Check 3: \(36 ÷ 3 = 12\). Check 4: \(36 ÷ 4 = 9\). Check 6: \(36 ÷ 6 = 6\). All of these leave no remainder, so they are factors.

Step 5 - Identify the Largest Factor

Since each factor pairs with another factor (i.e., 1 pairs with 36, 2 pairs with 18, etc.), once you reach the square root of 36 (which is 6), all pairs will have been found.

Step 6 - List All Factors

The complete list of factors includes both the smaller factors and their corresponding pairs: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Key Concepts

finding factors

To understand what factors are, think of them as the building blocks of a number. When finding factors, you are looking for all the integers that multiply together to give the original number. For example, the factors of 36 are numbers that, when multiplied by another number, result in 36. Start with the smallest number, which is 1, and test each consecutive number to see if it divides 36 without leaving a remainder. This helps ensure you don't miss any factors.

divisibility

Divisibility plays a key role in finding factors. A number is divisible by another if, when you divide them, the result is a whole number with no remainder. For 36, you test each number like this:
• 36 ÷ 1 = 36 (no remainder, so 1 is a factor)
• 36 ÷ 2 = 18 (no remainder, so 2 is a factor)
• 36 ÷ 3 = 12 (no remainder, so 3 is a factor)
Continue testing sequential numbers until you reach the square root of the number. Any numbers that divide 36 evenly without a remainder are factors.

pairs of factors

Factors usually come in pairs. This means for every factor of the number, there is another number that pairs with it to give the original number. For example, with 36, once you've identified that 1 is a factor, its pair is 36 because 1 × 36 = 36. Similarly, the factor 2 pairs with 18, since 2 × 18 = 36. Listing these pairs helps ensure all factors are found:
• 1 and 36
• 2 and 18
• 3 and 12
• 4 and 9
• 6 and 6
When you keep track of these pairs, it also helps to avoid repeating factors.

even division

When finding factors, you're looking for numbers that can divide the original number evenly. This means there is no remainder left after division. For example, dividing 36 by 4 gives an even result with no fractions left over:
• 36 ÷ 4 = 9 (even division, so 4 is a factor)
If there is a remainder, like 36 ÷ 5 = 7.2, the division is not even, so 5 is not a factor of 36. Always check for even division to accurately list all factors.