Question

Find the greatest common factor of the numbers. 15 and 45

Short answer

The greatest common factor of 15 and 45 is 15.

Step-by-step solution

Understand the task

The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into each of the numbers. This is what needs to be discovered for 15 and 45.

List all the Factors of Both Numbers

The factors for 15 are: 1, 3, 5 and 15. The factors for 45 are: 1, 3, 5, 9, 15, and 45.

Compare Both Lists and Find the Largest Common Factor

When we look at both lists, we see that 15 is the largest number in both lists, hence, it is the GCF of 15 and 45.

Key concepts

Factors of Numbers

Understanding the factors of numbers is crucial when solving mathematical problems involving divisibility and finding the greatest common factor (GCF). Factors are the whole numbers that can be multiplied together to produce a target number. For example, when we consider the number 15, we find that it can be created by multiplying 1 by 15, or 3 by 5. Hence, the factors of 15 are 1, 3, 5, and 15.

To list the factors of a number, a reliable method is to start from 1 and the number itself, and test divisibility by integers that lie between these two numbers. It's worth noting that factors always come in pairs unless the number is a perfect square. The process of identifying factors is foundational for many mathematical concepts and operations, such as determining the GCF.

Divisibility

Divisibility refers to the ability of one whole number to be divided by another without leaving a remainder. It's an essential concept for understanding how numbers relate to each other in a multiplicative context.

For instance, a number is divisible by another if the result of the division is an integer. In other words, if we divide two numbers and the remainder is zero, we can say that the first number is divisible by the second. This understanding of divisibility is particularly important when we are identifying factors of a number or solving for the GCF. Several tests for divisibility by common factors—like checking if the last digit is even for divisibility by 2 or if the sum of digits is divisible by 3 to determine divisibility by 3—can streamline the process when working with larger numbers.

GCF Calculation

The calculation of the Greatest Common Factor (GCF) is an important skill that helps students find the largest factor shared between two or more numbers. The GCF is particularly useful in simplifying fractions or integers and in problem-solving situations where common denominators are needed.

The step-by-step process to calculate the GCF begins by listing out all factors of the numbers in question, then identifying the common factors and finally selecting the largest of these common factors. For example, with the numbers 15 and 45, we list out their factors and see that the common factors are 1, 3, 5, and 15. Of these, 15 is the largest and therefore is the GCF of 15 and 45. In some cases, employing the Euclidean algorithm or prime factorization may provide a more efficient method of calculating the GCF, especially with larger numbers.