A
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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}
