The greatest common factor for the number 60 is 60.
Understanding the "greatest common factor" (GCF) for a single number like 60 often refers to its greatest factor, or equivalently, the greatest common factor of 60 with itself. In this context, the number itself is the largest value that can divide it exactly.
Understanding Factors of a Number
A factor of a number is an integer that divides the number evenly, leaving no remainder. To find the greatest common factor of 60, we first list all the numbers that divide 60 without leaving a remainder.
Factors of 60
The factors of 60 are the numbers that can multiply together to get 60, or numbers by which 60 can be divided exactly.
| Divisor | Result |
|---|---|
| 1 | 60 |
| 2 | 30 |
| 3 | 20 |
| 4 | 15 |
| 5 | 12 |
| 6 | 10 |
| 10 | 6 |
| 12 | 5 |
| 15 | 4 |
| 20 | 3 |
| 30 | 2 |
| 60 | 1 |
Therefore, the complete list of factors for 60 is: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF), sometimes called the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. When the question asks for the GCF "for" a single number, it implicitly refers to the GCF of that number with itself.
For example:
- GCF(12, 18): Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6. The greatest among these is 6. So, GCF(12, 18) = 6.
- GCF(60, 60): Both sets of factors are the same: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The greatest number common to both lists is 60.
Thus, the greatest common factor for 60 (or GCF of 60 and 60) is simply 60. It is the largest number in the list of its own factors.
Methods to Find GCF
While for a single number the GCF is the number itself, understanding how GCF is typically found for multiple numbers can be helpful.
- Listing Factors: As demonstrated above, list all factors of each number and identify the largest factor they share.
- Prime Factorization: Break down each number into its prime factors. The GCF is the product of the common prime factors raised to the lowest power they appear in any of the numbers. For 60, its prime factorization is $2^2 \times 3^1 \times 5^1$. If comparing GCF(60, 60), the common prime factors are identical, leading back to 60.
For more information on factors and GCF, you can explore resources like Khan Academy's explanation of factors and multiples.
Practical Insights
Understanding factors and GCF is fundamental in many mathematical concepts, including:
- Simplifying Fractions: Finding the GCF of the numerator and denominator allows you to reduce a fraction to its simplest form.
- Algebraic Expressions: Factoring out the GCF from terms in an expression.
- Problem Solving: Dividing items into equal groups or finding the largest possible size for an arrangement.
In summary, when asked for the greatest common factor of a single number, it refers to the largest number that divides it evenly, which is always the number itself.
