Problem 12 - Basic College Mathematics

Question

List all the factors of each number. $$ 13 $$

Step-by-Step Solution

Answer

The factors of 13 are 1 and 13.

Step 1: Understand the Definition of Factors

Factors of a number are integers that divide the number exactly without leaving a remainder. For instance, if a factor divides 13 evenly, there will be no remainder.

Step 2: Find the Factors of 13

To find the factors of 13, begin by considering the smallest factor, which is always 1. Next, test integers greater than 1 by dividing them into 13 to see if they leave any remainder.

Step 3: Identify Divisors

Divide 13 by each integer starting from 1 up to 13. Specifically: - 13 ÷ 1 = 13 (No remainder)- 13 ÷ 2 = 6.5 (Not an integer, remainder present)- 13 ÷ 3 through 13 ÷ 12 (All divisions result in a remainder)- 13 ÷ 13 = 1 (No remainder)

Step 4: List the Factors

Based on the calculations, the divisors without remainders are 1 and 13. Hence, the factors of 13 are 1 and 13.

Key Concepts

Integer Division

Integer division is a process where one integer is divided by another, resulting in a quotient and possibly a remainder. For example, if we divide 13 by 2, it goes 6 times and leaves a remainder. This is written as 13 ÷ 2 = 6 R1.

To conduct integer division:

• Divide the numbers.
• Ignore decimals or fractions.
• Note the remainder, if any.

Integer division is key when finding factors, as factors divide the number exactly without a remainder.

Remainders

Remainders occur when an integer does not divide evenly into another. For instance, 13 ÷ 2 leaves a remainder of 1. This indicates that 2 is not a factor of 13 since the division isn’t exact.

A factor must divide a number completely, leaving no remainder. While checking for factors of 13, you would find that most divisions leave remainders, except for when divided by 1 and 13.

Mathematical Factors

Mathematical factors are numbers you can multiply together to obtain another number. For example, 1 and 13 are factors of 13 because 1 × 13 equals 13.

To determine factors of a number:

• Start with 1 and the number itself.
• Test each number in between to see if they divide evenly.
• Only include numbers that divide without a remainder.

For 13, the only numbers that fulfill these criteria are 1 and 13, making them the factors of 13.