List all the prime factors of the following number: 198 (Hint: All the prime numbers that when multiplied together equal the number)
A. 2, 3, 11
B. 2, 3, 3, 11
C. 2,9,11
D. 2, 2, 3, 3, 11
A pipe has an inside diameter of 5/8 inch and a wall thickness of 3/16 inch. What is the outside diameter of the pipe?
A. 11/16 inch
B. 13/16 inch
C. 1 inch
D. 11/16 inch
Answer (2)
Find the prime factors of 198: 198 = 2 × 3 × 3 × 11 .
Calculate the outside diameter of the pipe: Outside Diameter = Inside Diameter + 2 * Wall Thickness.
Substitute the given values: Outside Diameter = 8 5 + 2 × 16 3 = 8 5 + 8 3 = 8 8 .
Simplify to find the outside diameter: 1 inch.
Explanation
Problem Analysis We are given two separate problems: finding the prime factors of 198 and finding the outside diameter of a pipe with a given inside diameter and wall thickness. We will address each problem separately.
Prime Factorization of 198 To find the prime factors of 198, we need to find all the prime numbers that, when multiplied together, equal 198. We can start by dividing 198 by the smallest prime number, 2: 198 ÷ 2 = 99 . Now we need to find the prime factors of 99. The smallest prime number that divides 99 is 3: 99 ÷ 3 = 33 . Again, 3 divides 33: 33 ÷ 3 = 11 . Finally, 11 is a prime number. So the prime factors of 198 are 2, 3, 3, and 11.
Understanding the Pipe Dimensions For the second problem, we are given that the inside diameter of the pipe is 8 5 inch and the wall thickness is 16 3 inch. To find the outside diameter, we need to add twice the wall thickness to the inside diameter. This is because the wall thickness is added to both sides of the pipe.
Setting up the Calculation The formula for the outside diameter is: Outside Diameter = Inside Diameter + 2 * Wall Thickness. Substituting the given values, we get: Outside Diameter = 8 5 + 2 × 16 3 .
Calculating the Outside Diameter Now, we simplify the expression: Outside Diameter = 8 5 + 16 6 . To add these fractions, we need a common denominator. The least common denominator of 8 and 16 is 16. So we convert 8 5 to a fraction with a denominator of 16: 8 5 = 8 × 2 5 × 2 = 16 10 . Now we can add the fractions: Outside Diameter = 16 10 + 16 6 = 16 16 = 1 inch.
Final Answer Therefore, the prime factors of 198 are 2, 3, 3, and 11, and the outside diameter of the pipe is 1 inch.
Examples
Prime factorization is used in cryptography to ensure secure data transmission. Pipe diameter calculations are essential in plumbing and construction to ensure proper fitting and flow.
The prime factors of 198 are 2, 3, 3, and 11, making the correct option B. For the pipe, the outside diameter calculated using the provided dimensions is 1 inch, corresponding to option C.
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