Expert: Paul Klarreich - 7/22/2006

Question
Which of the following (and how) cannot be factor of 2^n*3^k where n and k are positive integers:
a) 6
b) 8
c) 27
d) 42
e) 54

Answer
Hi, sweta,
 
Question:  Which of the following (and how) cannot be factor of 2^n*3^k where n and k are positive integers:
a) 6
b) 8
c) 27
d) 42
e) 54

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Any integer can be expressed as a product of primes in one and only one way. (The Fundamental Theorem of Arithmetic) If a number is a divisor of your expression:
2^n 3^k
then the only factors that appear in the prime factorization of that number can be 2's and 3's, and by saying n and k are positive integers, there must be at least one of each.

So:
6 = 2 3,  yes.
8 = 2^3,  NO. (no factor of 3)
27 = 3^3, NO. (no factor of 2)
42 = 2 3 7, NO.(has a prime factor other than 2 or 3)
54 = 2 3^3, yes.