a.) Of all the elements of D less than or equal to 20, which are D-prime and whi
ID: 1945293 • Letter: A
Question
a.) Of all the elements of D less than or equal to 20, which are D-prime and which are not?b.) Conjecture how to finish this statement: "Let n be an integer in D. Then n is D-prime if and only if..." PROVE YOUR CONJECTURE.
c.) Prove that every element in D can be written as a product of D-primes.
d.) Find an element of D that can be written in two different ways as the product of D-primes. (once you've done that, you will have shown that D does not satisfy a "fundamental theorem of D-prime arithmetic").
Explanation / Answer
a) The D-primes are 2,6,10,14,18, 4=2*2,8=2*4,12=2*6,16=2*8, 20=2*10 are not D-prime b)Let n be an integer in D. Then n is D-prime if and only if n is not divisible by 4. This is equivalent with showing n is not D-prime if and only if n is divisible by 4 If n is divisible by 4, then n=4k, for some k, then n=2*2k, that is a product of two elements in D Conversely, If n is not D-prime, then n=ab, with a,b in D, a=2k, b=2l, for some positive integers k,l, so n=4kl c) let n in D. If n is D-prime, we are done If n is not D-prime, then n=4k=2*2k. 2 is a D-prime. If k is odd, then 2k is D-prime by b). So we are done If k is even, then k=2l, for some l and n=2*2*2l. We continue the process. Let 2^s be the highest power of 2 that divides n n=2*2*...*2m, where the product is s times and m odd is a "factorization" of n into D-primes d)60=2*30=6*10 2,30,6,10 are D-primes
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