Question 1

What is GCF (Greatest Common Factor)?

Accepted Answer

The Greatest Common Factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. For example, the GCF of 24 and 36 is 12, because 12 is the largest number that divides both 24 and 36 evenly. GCF is also known as GCD (Greatest Common Divisor) or HCF (Highest Common Factor).

Question 2

How do you find GCF using the Euclidean algorithm?

Accepted Answer

The Euclidean algorithm is an efficient method to find GCF: 1) Divide the larger number by the smaller number, 2) Find the remainder, 3) Replace the larger number with the smaller number and the smaller number with the remainder, 4) Repeat until the remainder is 0, 5) The last non-zero remainder is the GCF. For example, GCF(24, 36): 36 = 24 × 1 + 12, 24 = 12 × 2 + 0, so GCF = 12.

Question 3

How do you find GCF using prime factorization?

Accepted Answer

To find GCF using prime factorization: 1) Find the prime factors of each number, 2) Identify the common prime factors, 3) Multiply the common prime factors to get the GCF. For example, 24 = 2³ × 3¹ and 36 = 2² × 3², so common factors are 2² × 3¹ = 12. This method is useful for understanding the mathematical structure of numbers.

Question 4

What is the relationship between GCF and LCM?

Accepted Answer

GCF and LCM are related by the formula: GCF(a, b) × LCM(a, b) = a × b. This means if you know the GCF of two numbers, you can find their LCM by dividing the product of the numbers by the GCF. For example, if GCF(24, 36) = 12, then LCM(24, 36) = (24 × 36) / 12 = 72. This relationship is fundamental in number theory.

Question 5

How do you find GCF of three or more numbers?

Accepted Answer

To find GCF of three or more numbers: 1) Find the GCF of the first two numbers, 2) Find the GCF of the result with the third number, 3) Continue this process for all remaining numbers. For example, GCF(12, 18, 24): First find GCF(12, 18) = 6, then find GCF(6, 24) = 6, so GCF(12, 18, 24) = 6.

Question 6

What are the applications of GCF in real life?

Accepted Answer

GCF has many practical applications: 1) Simplifying fractions (divide numerator and denominator by their GCF), 2) Finding equivalent ratios, 3) Solving problems involving equal distribution, 4) Calculating scale factors in engineering, 5) Optimizing resource allocation, 6) Solving word problems involving groups or sets, 7) Cryptography and number theory applications.

Question 7

How do you use GCF to simplify fractions?

Accepted Answer

To simplify a fraction using GCF: 1) Find the GCF of the numerator and denominator, 2) Divide both the numerator and denominator by the GCF, 3) The result is the simplified fraction. For example, to simplify 24/36: GCF(24, 36) = 12, so 24/36 = (24 ÷ 12)/(36 ÷ 12) = 2/3. This gives the fraction in its lowest terms.

Question 8

What are common factors and how do they relate to GCF?

Accepted Answer

Common factors are all the numbers that divide two or more numbers evenly. The GCF is the largest of these common factors. For example, the common factors of 24 and 36 are: 1, 2, 3, 4, 6, 12. The GCF is 12, which is the largest common factor. Finding all common factors can help understand the relationship between numbers.

Question 9

How do you find GCF when numbers are large?

Accepted Answer

For large numbers, the Euclidean algorithm is more efficient than listing factors: 1) Use the Euclidean algorithm to find GCF, 2) This method works well even with very large numbers, 3) The algorithm reduces the problem size quickly, 4) It's computationally efficient and doesn't require finding all factors. For example, GCF(123456, 789012) can be found efficiently using the Euclidean algorithm.

Question 10

What are the properties of GCF?

Accepted Answer

GCF has several important properties: 1) GCF(a, b) = GCF(b, a) (commutative), 2) GCF(a, a) = a, 3) GCF(a, 0) = a, 4) GCF(a, 1) = 1, 5) If a divides b, then GCF(a, b) = a, 6) GCF(ka, kb) = k × GCF(a, b) for any positive integer k, 7) GCF(a, b) = GCF(a, b - ka) for any integer k. These properties are useful in mathematical proofs and calculations.

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