Problem Solving

lheiannie07 wrote:The "primeness" of a positive integer x is defined as the positive difference between its largest and smallest prime factors. Which of the following has the greatest primeness?

A) 10
B) 12
C) 14
D) 15
E) 18

How will i start to solve this? Can any experts help me?

OA C

A) 10:

Factors of 10 are: 1, 2, 4, 5, and 10; the largest prime factor = 5 and the smallest prime factor = 2; thus, the difference between the largest and the smallest prime factors = 5 - 2 = 3.

B) 12:

Factors of 12 are: 1, 2, 3, 4, 6, and 12; the largest prime factor = 3 and the smallest prime factor = 2; thus, the difference between the largest and the smallest prime factors = 3 - 2 = 1.

C) 14:

Factors of 14 are: 1, 2, 7, and 14; the largest prime factor = 7 and the smallest prime factor = 2; thus, the difference between the largest and the smallest prime factors = 7 - 2 = 5.

D) 15:

Factors of 15 are: 1, 3, 5, and 15; the largest prime factor = 5 and the smallest prime factor = 3; thus, the difference between the largest and the smallest prime factors = 5 - 3 = 2.

E) 18:

Factors of 18 are: 1, 2, 3, 6, 9, and 18; the largest prime factor = 3 and the smallest prime factor = 2; thus, the difference between the largest and the smallest prime factors = 3 - 2 = 1.

The greatest difference is 5 for number 14 (option C).

The correct answer: C

Hope this helps!

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Hi lheiannie07,

This question literally tells you to use prime factorization on the each of the 5 answer choices - as we're looking for the answer with the largest DIFFERENCE between its largest and smallest prime factors.

Answer A: 10 = (2)(5)
Answer B: 12 = (2)(2)(3)
Answer C: 14 = (2)(7)
Answer D: 15 = (3)(5)
Answer E: 18 = (2)(3)(3)

Looking at these options, the largest difference isn't hard to spot...

Final Answer: C

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Rich