The greatest common factor (GCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. In order to find the GCF of 24 and 40, we must first find the factors of both numbers and then identify the largest factor that they have in common.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. We can find these factors by dividing 24 by each of the integers from 1 to 24 and noting which ones result in a whole number. For example, 24 divided by 3 is 8, which is a whole number, so 3 is a factor of 24. Similarly, 24 divided by 6 is 4, which is also a whole number, so 6 is also a factor of 24.
The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. We can find these factors in the same way we found the factors of 24.
To find the GCF of 24 and 40, we need to identify the largest factor that they have in common. Looking at the two lists of factors, we can see that both 24 and 40 have 1, 2, 4, 8, and as common factors. However, 10, 20, and 40 are factors of 40 that are not factors of 24.
Therefore, the largest factor that 24 and 40 have in common is 8. This means that 8 is the GCF of 24 and 40.
We can also check this answer by dividing both 24 and 40 by 8 and confirming that there is no remainder. 24 divided by 8 is 3 and 40 divided by 8 is 5. Since both of these are whole numbers, we know that 8 is a common factor of 24 and 40.
It is worth noting that the GCF of two numbers can also be found using prime factorization. This involves breaking each number down into its prime factors and then identifying the factors that they have in common. For example, the prime factorization of 24 is 2 x 2 x 2 x 3 and the prime factorization of 40 is 2 x 2 x 2 x 5. The factors that they have in common are 2 x 2 x 2, which equals 8.
In conclusion, the greatest common factor of 24 and 40 is 8. This means that 8 is the largest positive integer that divides both 24 and 40 without leaving a remainder.