Let 2N 2 4 6 be the set of even natural numbers and call a

Let 2N = {2, 4, .6, ...} be the set of even natural numbers and call a number e epsilon 2N a \"Liebeck number\" if e cannot be expressed as a product of two other members of 2N. Which of {2.4.6.8.10} are Liebeck numbers? What do you get if you write a Liebeck number as a product of primes (in N)? Show that every clement of 2N can be expressed as a product of Liebeck numbers. Is it true that every element of 2N can be expressed uniquely as a product of Liebeck numbers?

Solution

3. (a) A number e 2N is said to be a “Liebeck number” if e cannot be expressed as a product of two other numbers of 2N. Since any number of 2N is of the form 2n for some natural number n, product of any two numbers of 2N is of the form (2n)(2m) = 4nm for some natural numbers n, m. Therefore the product must be divisible by 4. So, Liebeck numbers must not be divisible by 4.

Therefore, for the set {2,4,6,8,10} Liebeck numbers are 2, 6, and 10 as these numbers are not divisible by 4.

(b) A Liebeck number which can be expressed as a product of primes (in N) is of the form 2p where p is a odd prime (note that 2 is a prime number).

(c) Any Liebeck number is of the form 2x where x is some odd positive integer. Therefore, product of two distinct Liebeck numbers is of the form (2x)(2y) for some odd positive integer x, y; x y

(2x)(2y) = 2(2xy) = 2n, where n = 2xy N

As, 2n 2N, (2x)(2y) 2N

Therefore, we can say that every element of 2N can be expressed as a product of Liebeck numbers.

(d) This is not true as 60 2N, 60 can be expressed in two different ways:

60 = 6 x 10

60 = 2 x 30

2, 6, 10, 30 are Liebeck numbers.

Therefore, we can conclude that every element of 2N cannot be expressed uniquely as a product of Liebeck numbers. (Answer)