Name three prime factors of each of the following products

Name three prime factors of each of the following products: a. 3 x 7³ x 22 b. 27 x 22 la c. 294 × 116 x 25

The Correct Answer and Explanation is :

Let’s begin by factoring each product and identifying three prime factors.

Part a: ( 3 \times 7^3 \times 22 )

We start by factoring each number into prime factors:

• ( 3 ) is a prime number.
• ( 7^3 ) means ( 7 \times 7 \times 7 ), so the prime factor is ( 7 ).
• ( 22 = 2 \times 11 ), so the prime factors are ( 2 ) and ( 11 ).

Thus, the prime factors of the product ( 3 \times 7^3 \times 22 ) are:

• 3 (prime)
• 7 (prime, repeated three times)
• 2 (prime)
• 11 (prime)

Three prime factors: ( 3, 7, 2 ) (note that 7 is repeated but counts as a prime factor).

Part b: ( 27 \times 22 )

Now, we factor each term:

• ( 27 = 3^3 ), so the prime factor is ( 3 ).
• ( 22 = 2 \times 11 ), so the prime factors are ( 2 ) and ( 11 ).

Thus, the prime factors of the product ( 27 \times 22 ) are:

• 3 (prime, repeated three times)
• 2 (prime)
• 11 (prime)

Three prime factors: ( 3, 2, 11 ).

Part c: ( 294 \times 116 \times 25 )

We factor each number:

• ( 294 = 2 \times 3 \times 7^2 ), so the prime factors are ( 2, 3, 7 ).
• ( 116 = 2^2 \times 29 ), so the prime factors are ( 2 ) and ( 29 ).
• ( 25 = 5^2 ), so the prime factor is ( 5 ).

Thus, the prime factors of the product ( 294 \times 116 \times 25 ) are:

• 2 (prime, repeated multiple times)
• 3 (prime)
• 7 (prime)
• 29 (prime)
• 5 (prime)

Three prime factors: ( 2, 3, 5 ).

Explanation of Prime Factorization:

Prime factorization is the process of expressing a number as the product of prime numbers. Every composite number can be factored into prime numbers, which are the building blocks of all numbers. Prime factors are crucial in many areas of mathematics, particularly in number theory, since they provide insight into the divisibility and structure of numbers. When factoring large expressions, it’s helpful to break down each component into its prime factors to simplify calculations or to solve problems like finding the greatest common divisor or least common multiple.