Question

Find the greatest common factor. $$49,64$$

Short answer

The greatest common factor of 49 and 64 is \(1\).

Step-by-step solution

Finding Factors of 49

First, find all the factors of 49. The factors of 49 are \(1, 7\), and \(49\).

Finding Factors of 64

Next, find all the factors of 64. The factors of 64 are \(1, 2, 4, 8, 16, 32\), and \(64\).

Identifying the Common Factors

Now, identify the common factors between the two lists. The only common factor is \(1\).

Key concepts

Finding Factors

When solving mathematical problems, particularly those related to number theory or algebra, 'finding factors' is a fundamental skill. A factor of a number is an integer which divides that number without leaving a remainder. For instance, when finding factors of a number like 49, one approach is systematically testing each integer to see which numbers divide 49 evenly.

To do this effectively, remember that factors come in pairs with one number being small and its corresponding factor being larger. For the number 49, you start with 1, which is a universal factor for any integer, and pair it with 49 itself (since 1 x 49 = 49). Moving up, you soon realise that 7 is a factor because 7 x 7 also equals 49. No other numbers divide 49 evenly, leaving us with the factors 1 and 7.

Finding factors is a crucial step in identifying the greatest common factor, simplifying fractions, and solving various equation types. It's the building block for understanding more complex algebraic operations.

Common Factors

Once the factors of individual numbers are identified, the next step is to find 'common factors'. These are numbers that are factors of two or more given integers. Common factors are important in finding the greatest common factor (GCF) of a set of numbers, which is the largest number that divides all the numbers in the set without any remainder.

For example, in finding common factors for 49 and 64, you'll list the factors for each number—1, 7, and 49 for the former, and 1, 2, 4, 8, 16, 32, and 64 for the latter. The next step involves comparing these lists to see which factors appear in both. In our case, the number 1 is the only common factor. It's worth noting that 1 is a common factor for any set of integers, being the most fundamental multiplicative identity.

Familiarity with common factors helps in simplifying algebraic expressions and finding the simplest form of fractions—a key concept in intermediate mathematics.

Factorization

The process of breaking down a number into its factors is known as 'factorization'. In its simplest form, this means representing a number as a product of its factors. This concept becomes particularly valuable when dealing with composite numbers, which have more factors than just 1 and themselves.

There are different methods of factorization, but one commonly used technique is prime factorization, where you break a number down into its prime number constituents. For example, while 64 is not a prime, it can be expressed as the product of primes: 64 = 2 x 2 x 2 x 2 x 2 x 2, or equivalently, 64 = 2^6.

Having a solid grasp on factorization is essential for finding the greatest common factor, solving quadratic equations, and simplifying higher-level mathematical expressions. It allows students to deconstruct complex numerical relationships into more manageable parts, providing an essential tool for deeper comprehension of mathematical concepts.