I have two full questions on some number theory questions I've been working on, I guess my best bet would be to post them separately.(adsbygoogle = window.adsbygoogle || []).push({});

1) Suppose that n is in N (natural numbers), p_{1},....,p_{n}are distinct primes, and l_{1},....l_{n}are nonnegative integers. Let m = p_{1}^{l1}p_{2}^{l2}....p_{n}^{ln}. Let d be in N such that d ≥ 2 and d divides m.

a) Using the fundamental theorem of arithemetic prove that p is a prime that divides d, then p_{i}for some i in {1,...,n}.

Attempt: Since d is in N and d ≥ 2, this means that d itself is a product of primes (prime factorization) or a prime itself. This means d = p_{1}p_{2}...p_{n}(product of primes). Then there exists a p that divides d. (Since we are starting from 1 in the natural numbers)

Is this the right idea and what would be a clearer way of writing this as a proof?

b) Suppose the i is in {1,...,n}. Prove that p_{i}^{k}does not divide d if

k > l_{i}

Attempt: if k > l_{i}then d ≠ p_{i}^{k}, but d may equal any prime in m = p_{1}^{l1}p_{2}^{l2}....p_{n}^{ln}since we know d | m.

this would imply d has a cononical factorization : p_{1}^{l1}p_{2}^{l2}....p_{n}^{ln}and we know k > l_{i}therefore p_{i}^{k}does not divide d.

Right idea?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Number Theory fundamental theorem of arithemetic

**Physics Forums | Science Articles, Homework Help, Discussion**