# Solutions to Questions on Prime Factorization

Detailed solutions to the questions on Prime Factorization are presented.

 A Prime Factors Calculator may be used to check your answers. Answer the following questions on prime factorization. Which of the following is not a prime factorization? a) 20 = 2×10 , b) 14 = 2×7 , c) 64 = 4 3 , d) 120 = 2 3 × 15 Solution Prime factorization involves only prime numbers. Hence The following are not prime factorization. a) 20 = 2×10 ; 10 is not a prime number. c) 64 = 4 3 ; 4 is not a prime number. What is the prime factorization of the following numbers? a) 28 , b) 32 , c) 100 , d) 126 , e) 900 Solution Prime factorization of the above are. a) 28 = 2 × 2 × 7 = 2 2 × 7 b) 32 = 2 × 2 × 2 × 2 × 2 = 2 5 c) 100 = 2 × 2 × 5 × 5 = 2 2 × 5 2 d) 126 = 2 × 3 × 3 × 7 = 2 × 3 2 × 7 e) 900 = 2 × 2 × 3 × 3 × 5 × 5 = 2 2 3 2 5 2 Two numbers A and B are given by: A = 2 3 × 5 2 × 11 and B = 2 2 × 5 × 13. What is the prime factorization of A×B? Solution Prime factorization. A×B = (2 3 × 5 2 × 11 ) × (2 2 × 5 × 13) = 2 × 2 × 2 × 5 × 5 × 11 × 2 × 2 × 5 × 13 = 2 5 × 5 3 × 11 × 13 Find the prime factorization of 100 and 70 and then the prime factorization of 7000 knowing that 7000 = 100 × 70 Solution Prime factorization of 100 and 70. 100 = 2 2 × 5 2 70 = 2 × 5 × 7 Use the fact that 7000 = 100 × 70 to find the prime factorization 7000. 7000 = 100 × 70 = (2 2 × 5 2) × (2 × 5 × 7) = 2 × 2 × 5 × 5 × 2 × 5 × 7 = 2 3 × 5 3 × 7 a) Find the prime factorization of 10. b) Use the result in part a) and the fact that 100 = 10 × 10 to find the prime factorization of 100. c) Use the result in part a) and the fact that 1000 = 10 × 10× 10 to find the prime factorization of 1000. d) Use the results in parts a), b) and c) to find a pattern of prime factorization and find the prime factorization of 1000,000. Solution a) Prime factorization of 10. 10 = 2× 5 b) Prime factorization of 100. 100 = 10 × 10 = 10 2 = (2× 5) × (2× 5) = 2 2 5 2 c) Prime factorization of 1000. 1000 = 10 × 10 × 10 = 10 3 = (2× 5) × (2× 5) × (2× 5) = 2 3 5 3 d) Prime factorization of 1000,000 using the above pattern. 1000,000 = 10 6 = 2 6 5 6