Solutions to Questions on Prime Factorization

Detailed solutions to the questions on Prime Factorization are presented.

Answer the following questions on prime factorization.

1. 
   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.
   
2. 
   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
   
3. 
   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
   
4. 
   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
   
5. 
   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

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