Problem 10 - Basic College Mathematics

Question

List all the factors of each number. $$ 9 $$

Step-by-Step Solution

Answer

The factors of 9 are 1, 3, and 9.

Step 1 - Understand the Definition of Factors

A factor of a number is an integer that divides the number without leaving a remainder. To find all the factors of 9, we need to identify all integers that can multiply together to give 9.

Step 2 - Start with the Smallest Integers

Begin by checking the smallest integers: 1, 2, and so forth. Since 1 is a factor of every number, we start with 1. 1 divides 9 and leaves no remainder, so it is a factor.

Step 3 - Check for Division by Larger Numbers

Next, check 2. Since 9 divided by 2 is not an integer (it leaves a remainder), 2 is not a factor. Now check 3. Since 3 * 3 = 9, 3 is a factor. Continue this process.

Step 4 - List All Found Factors

After checking all numbers up to 9, we find that the numbers 1, 3, and 9 divide 9 completely. Therefore, the factors of 9 are 1, 3, and 9.

Step 5 - Review the Complete List

Review the list of factors: it should include only numbers that divide 9 without leaving a remainder. So, the complete list of factors for 9 is {1, 3, 9}.

Key Concepts

Factors

Factors are fundamental in mathematics as they help us understand the composition of numbers. When we talk about factors, we refer to integers that can divide a given number without leaving a remainder.
For example, take the number 9. We need to find all its factors. To do this, we will check each integer to see if it divides 9 completely:

• First, 1 is a factor of every number. Since 1 divides 9 and leaves no remainder, it's a factor.
• Next is 2. If you divide 9 by 2, you get 4.5, which is not an integer, meaning 2 leaves a remainder.
• Then, we try 3. The calculation is straightforward: 3 multiplied by 3 gives 9. So, 3 is a factor.
• Lastly, 9 itself is a factor because 9 divided by 9 is 1, an integer without remainder.

After checking, we conclude that the factors of 9 are 1, 3, and 9. It is essential to thoroughly verify your steps to ensure you capture all factors correctly.

Integer Division

Integer division is another crucial concept to understand when dealing with factors. It refers to the division of one integer by another where the result is also an integer, without any fraction or decimal.
Let's apply this to our example. When we divide 9 by 3, the exact quotient is 3—an integer. As there are no leftover parts (no decimal or fraction), we can say that 9 is completely divisible by 3. This is a clear indication that 3 is a factor of 9.
On the other hand, if we divide 9 by 4, the result is 2.25, which is not an integer. It shows that 4 cannot divide 9 into whole numbers, hence not a factor.
Understanding this helps us identify factors. We are always looking for exact division results, ensuring no remainders are involved.

Remainder

Understanding the concept of remainders is essential in factorization. A remainder represents the leftover part of an integer division that doesn’t fit into whole numbers.
For instance, when 9 is divided by 2, the quotient is 4 with a remainder of 1 (since 2 times 4 is 8, and adding the remainder 1 equals 9). This leftover part—1 in this case— indicates that 2 isn’t a factor of 9.
Conversely, when dividing 9 by 3, the quotient is 3 with no remainder. It shows a perfect division without leftovers, meaning 3 is a factor of 9.
In summary, when finding factors, any division resulting in a remainder means the divisor is not a factor. Only those integers that divide the number without any remainder can be considered factors.