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In mathematics, a factor is a number that divides another number evenly, meaning there is no remainder left over. When we talk about factors, we're usually referring to positive integers. Finding the factors of a number involves determining all the numbers that can multiply together to form that number. For example, if you consider the number 6, its factors are 1, 2, 3, and 6 because each of these numbers can be multiplied by another number to give 6 (i.e., 1 × 6, 2 × 3).
Factors are useful in various areas of mathematics, especially when simplifying fractions or finding greatest common divisors. To find the factors of a number like 78, you would list all numbers that divide 78 without leaving a remainder. Similarly, when you find factors for 105, you follow the same process. Identifying these factors is essential when finding common factors of two numbers, which are important in solving problems that involve both numbers together.

Divisibility is a fundamental concept that describes whether one number can be divided by another without leaving a remainder. Understanding divisibility helps in identifying factors and common factors between numbers. For a number to be considered divisible by another, the division must result in a whole number.
There are simple rules that can aid in determining the divisibility of numbers. For instance:

• A number is divisible by 2 if its last digit is even.
• A number is divisible by 3 if the sum of its digits is a multiple of 3.
• A number is divisible by 5 if its last digit is 0 or 5.

Applying these divisibility rules can quickly help determine the factors of a number. For example, to find if 105 is divisible by 3, you sum its digits: 1 + 0 + 5 = 6, and since 6 is a multiple of 3, 105 is divisible by 3.

When dealing with pairs of numbers, it's common to look for their common factors. A pair of numbers refers simply to two numbers considered together, such as 78 and 105. Finding common factors involves listing all the factors of each number and identifying the numbers they have in common.
The process can be broken down as follows:

• First, determine all factors of the first number.
• Then, list all factors of the second number.
• Finally, compare the two lists and identify the numbers present in both lists.

In our example, the common factors of 78 and 105 were found to be 1 and 3. These are the numbers that can evenly divide both 78 and 105. Understanding common factors helps in simplifying ratios and fractions, which can be especially useful in solving algebraic equations and problems involving proportions.