Question 1

What is GCF (Greatest Common Factor)?

Accepted Answer

The GCF, or Greatest Common Factor, is the largest number that divides two or more numbers without leaving a remainder. For example, the GCF of 24 and 36 is 12, since 12 is the largest number that divides both 24 and 36 evenly. The GCF is also called the GCD (Greatest Common Divisor).

Question 2

How do you find the GCF of two or more numbers?

Accepted Answer

To find the GCF: 1) List all factors of each number. 2) Identify common factors. 3) Choose the largest common factor. Method 2 - Euclidean Algorithm: Repeatedly divide larger by smaller until remainder is zero. The last non-zero divisor is the GCF. Method 3 - Prime Factorization: Write each number as product of primes, multiply common primes to lowest power.

Question 3

What is the Euclidean algorithm for finding GCF?

Accepted Answer

The Euclidean algorithm efficiently finds GCF: 1) Divide the larger number by the smaller. 2) Replace the larger with the smaller. 3) Replace the smaller with the remainder. 4) Repeat until remainder is 0. The last non-zero remainder is the GCF. Example: GCF(48,18): 48÷18=2R12, 18÷12=1R6, 12÷6=2R0, so GCF=6.

Question 4

What is the difference between GCF and LCM?

Accepted Answer

GCF (Greatest Common Factor) is the largest number that divides all given numbers. LCM (Least Common Multiple) is the smallest number that is a multiple of all given numbers. For 12 and 18: GCF = 6, LCM = 36. They are related: GCF(a,b) × LCM(a,b) = a × b.

Question 5

When is the GCF equal to 1?

Accepted Answer

The GCF equals 1 when two or more numbers have no common factors except 1. Such numbers are called relatively prime or coprime. Examples: 7 and 11 have GCF=1, 15 and 22 have GCF=1. Prime numbers are always relatively prime to each other.

Question 6

How do you find the GCF using prime factorization?

Accepted Answer

Using prime factorization: 1) Factor each number into primes. 2) List common primes with lowest exponents. 3) Multiply these primes. Example: 24 = 2³×3, 36 = 2²×3². Common primes: 2² and 3¹. GCF = 2²×3 = 4×3 = 12.

Question 7

What are practical applications of GCF?

Accepted Answer

GCF is used in: simplifying fractions (divide numerator and denominator by GCF), algebraic factoring (factor out GCF from terms), equal distribution problems, tile/lattice patterns, cryptography, and scheduling problems. For example, to simplify 24/36, divide by GCF(24,36)=12 to get 2/3.

Question 8

Can you find the GCF of more than two numbers?

Accepted Answer

Yes, to find GCF of multiple numbers: 1) Find GCF of first two numbers, 2) Find GCF of that result and the next number, 3) Repeat for all numbers. Example: GCF(12,18,24) = GCF(GCF(12,18),24) = GCF(6,24) = 6. The final result divides all original numbers.

Question 9

What is the GCF of prime numbers?

Accepted Answer

If two prime numbers are the same, their GCF is that prime. If different primes, their GCF is 1. Example: GCF(7,7) = 7, GCF(5,13) = 1. This is because prime numbers have only two factors: 1 and itself, so they share only the factor 1.

Question 10

How does the GCF relate to simplifying algebraic expressions?

Accepted Answer

In algebra, finding the GCF of coefficients and variables helps factor and simplify expressions. Example: In 6x² + 12x, the GCF is 6x. Factoring gives 6x(x + 2). This is used in solving equations, simplifying radicals, and polynomial factorization.

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