Greatest Common Factor of 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000 by Prime Factorization

Given Numbers: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000
Identify the Prime Factors of 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000 by Prime Factorization

• Prime Factorization of 100: 100 = 2 x 2 x 5 x 5 = 2² x 5²
• Prime Factorization of 200: 200 = 2 x 2 x 2 x 5 x 5 = 2³ x 5²
• Prime Factorization of 300: 300 = 2 x 2 x 3 x 5 x 5 = 2² x 3¹ x 5²
• Prime Factorization of 400: 400 = 2 x 2 x 2 x 2 x 5 x 5 = 2⁴ x 5²
• Prime Factorization of 500: 500 = 2 x 2 x 5 x 5 x 5 = 2² x 5³
• Prime Factorization of 600: 600 = 2 x 2 x 2 x 3 x 5 x 5 = 2³ x 3¹ x 5²
• Prime Factorization of 700: 700 = 2 x 2 x 5 x 5 x 7 = 2² x 5² x 7¹
• Prime Factorization of 800: 800 = 2 x 2 x 2 x 2 x 2 x 5 x 5 = 2⁵ x 5²
• Prime Factorization of 900: 900 = 2 x 2 x 3 x 3 x 5 x 5 = 2² x 3² x 5²
• Prime Factorization of 1,000: 1,000 = 2 x 2 x 2 x 5 x 5 x 5 = 2³ x 5³

Determine the Prime Factor(s) that are common to the Prime Factorization of each number:

• 2 and 5 are Prime Factors common to the Prime Factorizations of 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000

Determine the least number of occurrences of 2 and 5 among the Prime Factorizations:

• 2 occurs at least 2 times in each Prime Factorization
• 5 occurs at least 2 times in each Prime Factorization

As a result, the Greatest Common Factor (GCF) of 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000 is equal to 2 x 2 x 5 x 5, which equals the product of 2 and 5 raised to the powers of their least number of occurrences or smallest exponent among the Prime Factorizations:

• GCF(100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000) = 2² x 5² = 2 x 2 x 5 x 5
• GCF(100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000) = 4 x 25
• GCF(100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000) = 100

Therefore, the Greatest Common Factor of 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000 by Prime Factorization = 100