# Thread: Fundamental Theorem of Arithmetic

1. ## Fundamental Theorem of Arithmetic

Use the Fundamental Theorem of Arithmetic to show every positive integer, n>0, can be written uniquely as the product of a power of 2 and an odd number.

2. Suppose that there are two distinct representations of the same number as the product of a power of 2 and an odd number (let us call them $2^{p_a}O_a$ and $2^{p_b}O_b$. This number has a unique decomposition into primes, from which it immediately follows that $p_a = p_b$, since $O_a$ and $O_b$ contain no factors of 2, and 2 is prime. Thus $2^{p_a} = 2^{p_b} = c$. Now $cO_a = cO_b$, and since $c \geq 1$, it follows that $O_a = O_b$, thus demonstrating that the two representations are not distinct, and hence the number has a unique representation as a power of 2 and an odd number.

3. I see where your proof shows uniqueness, but I don't see how it shows that the integers can be written as the product of a power of 2 and an odd number.

4. Originally Posted by tarheelborn
I see where your proof shows uniqueness, but I don't see how it shows that the integers can be written as the product of a power of 2 and an odd number.
Oh, I guess I just overlooked that part because it seemed obvious. Write a number as its unique product of primes:

$2^{p_1}3^{p_2}5^{p_3}...$

Now, using associativity, this is equal to

$2^{p_1} \cdot (3^{p_2}5^{p_3}...)$

The left term is a power of two and the right term is an odd number, because every product of only odd numbers is odd. You can prove that statement easily using modular arithmetic.

5. Thank you!!!