Question

Basic Definitions Define the terms given. Factor

Short answer

Answer: In mathematics, factors are numbers that divide another number (called the multiple) without leaving a remainder. To find the factors of a given number, start by listing the number 1 and the given number as factors. Test each integer greater than 1 to see if it divides the given number without leaving a remainder. If it divides evenly, add it to the list of factors and also add the quotient. Continue this process until you have tested all the integers up to the square root of the given number, as factors often come in pairs. Finally, list the factors in increasing order, removing any duplicates.

Step-by-step solution

Definition

In mathematics, a factor is a number that divides another number (called the multiple) without leaving a remainder. In other words, the multiple is the product of two or more factors. For example, the factors of the number 6 are 1, 2, 3, and 6 because these numbers can divide 6 without leaving any remainder.

Examples

Here are some examples to further illustrate the concept of factors: 1. The factors of 8 are 1, 2, 4, and 8 because 8 can be divided by these numbers without leaving a remainder (8 = 1 x 8, 8 = 2 x 4). 2. The factors of 15 are 1, 3, 5, and 15 because 15 can be divided by these numbers without leaving a remainder (15 = 1 x 15, 15 = 3 x 5). 3. The factors of 21 are 1, 3, 7, and 21 because 21 can be divided by these numbers without leaving a remainder (21 = 1 x 21, 21 = 3 x 7).

Common Misconceptions

A common misconception when dealing with factors is confusing them with multiples. Remember, a factor divides another number (the multiple) without leaving a remainder, whereas a multiple is the result of multiplying a number by any integer. For example, the multiples of 3 are 3, 6, 9, 12, 15, and so on, because these numbers can be created by multiplying 3 by an integer (3 x 1, 3 x 2, 3 x 3, etc.). Another misconception is to assume that the factors of a number always include numbers greater than the number itself. However, factors are always less than or equal to the number being factored, except when factoring 1 (1 has only one factor, which is itself).

Finding Factors

To find the factors of a given number: 1. Start by listing the number 1 and the given number as factors, since 1 and the number itself are always factors. 2. Test each integer greater than 1 to see if it divides the given number without leaving a remainder. 3. If it divides evenly, add it to the list of factors and also add the quotient. 4. Continue this process until you have tested all the integers up to the square root of the given number. Factors often come in pairs, so this will usually provide a complete list. 5. List the factors in increasing order, removing any duplicates. For example, to find the factors of 28: 1. List 1 and 28 as factors. 2. Test 2: 28 ÷ 2 = 14 (remainder 0). Add 2 and 14 to the list of factors. 3. Test 3: 28 ÷ 3 = 9 with a remainder. No new factors. 4. Test 4: 28 ÷ 4 = 7 (remainder 0). Add 4 and 7 to the list of factors. 5. Test 5: 28 ÷ 5 = 5 with a remainder. No new factors. 6. Since we've tested up to the square root of 28 (approximately 5.29), we have the complete list of factors: 1, 2, 4, 7, 14, and 28.

Key concepts

Divisibility in Mathematics

Understanding divisibility in mathematics is foundational for studying factors, multiples, and number relationships. Divisibility refers to the ability of one number to be divided by another without leaving a remainder. It's a simple yet powerful concept that allows us to determine whether a number is a factor of another.

For instance, we say that 10 is divisible by 2 because when 10 is divided by 2, the result is 5 with no remainder. However, 10 is not divisible by 3 as it leaves a remainder of 1. This concept is used extensively when simplifying fractions, factoring equations, and analyzing number patterns.

When checking for divisibility, certain rules make the process quicker, such as knowing that any even number is divisible by 2 or that if the sum of a number's digits is divisible by 3, the number itself is divisible by 3. These rules are helpful shortcuts in finding factors or testing for divisibility without performing long divisions.

Multiple and Factor Relationship

The relationship between multiples and factors is fundamental in understanding how numbers interact in division and multiplication. A multiple is the product of a number and any integer. Conversely, a factor is a number that divides another number, the multiple, cleanly without leaving a remainder.

Here's an easy way to visualize their relationship: if you think of multiplication as building a structure (like a tower with blocks), each block would be a factor that contributes to the height (the multiple) of the structure. If you have 4 blocks and stack them on top of each other, your tower would have a height of 4. In this case, 1 and 4 are factors of 4, and 4 is a multiple of 1 and 4.

Understanding this relationship is crucial for operations such as simplifying fractions or finding the least common multiple and greatest common factor. These concepts help us solve various mathematical problems and are the basis for more complex algebraic functions.

Finding Factors of a Number

When it comes to finding factors of a number, there's a clear method to follow that ensures you don't miss any potential factors. Factors of a number are the numbers that multiply together to produce the original number. They are the divisors that leave no remainder when you divide the original number by them.

Starting with the number 1 and the number itself is essential because these are guaranteed factors. To methodically find the rest, incrementally test each integer between 2 and the number's square root to see if it divides the number evenly. If it does, both the integer and the result (or quotient) are factors.

For example, to find factors of 12, you begin with 1 and 12. You would then check 2 (12 ÷ 2 = 6, so 2 and 6 are factors), proceed to 3 (12 ÷ 3 = 4, so 3 and 4 are factors), and stop there since 3 is close to the square root of 12. No need to check past 3 because factors beyond the square root would have already been found as quotients. The complete list of factors for 12, therefore, is 1, 2, 3, 4, 6, and 12.