Need to find a number according to this:

Hello :)

How I would have to solve it, what do you think?

I need to find the smallest natural number **X** [ http://latex.codecogs.com/gif.latex?...lon \mathbb{N}] with these features:

1) http://latex.codecogs.com/gif.latex?...} \frac{1}{2}x is a square of whole number

2) http://latex.codecogs.com/gif.latex?...} \frac{1}{3}x is a cube of whole nuber

3) http://latex.codecogs.com/gif.latex?...} \frac{1}{5}x is a fifth degree of whole number

*In the natural numbers, ***N, find the lowest such that half of it is a perfect square, one-third is a perfect cube, and one fifth is a perfect fifth power**

Re: Need to find a number according to this:

Quote:

Originally Posted by

**Sott**

Hi Sott! :)

Let's try a couple.

Is either of your 3 features satisfied if we pick x=1 (I'm assuming that your $\displaystyle \mathbb N$ starts with 1).

What about x=2?

And x=3, x=4, x=5?

Re: Need to find a number according to this:

You have to get a whole number.... Does 1/2 or 2/3 or 4/5 or 5/2 is a whole number? No...

Re: Need to find a number according to this:

Re: Need to find a number according to this:

Quote:

Originally Posted by

**Sott** You have to get a whole number.... Does 1/2 or 2/3 or 4/5 or 5/2 is a whole number? No...

Ah, it seems I misunderstood.

These features have to be true all at the same time.

Then 2 must divide x, 3 must divide x, and 5 must divide x.

Therefore x is of the form 2.3.5.(product of prime numbers).

After dividing by 2 all powers of the primes in the factorization must be even.

So x must be of the form 2^odd . 3^even . 5^even . (other primes)^even.

After dividing by 3 the powers must be multiples of 3.

So x must be of the form 2^(3k) . 3^(3l+1) . 5^(3m) . (other primes)^(multiple of 3).

After dividing by 5 the powers must be multiples of 5.

So x must be of the form 2^(5k') . 3^(5l') . 5^(5m'+1) . (other primes)^(multiple of 5).

To find the smallest number, we need the lowest powers.

And the numbers 2, 3, and 5 need a power of at least 1.

The smallest number that satisfies that is:

x = 2^15 . 3^10 . 5^6

Re: Need to find a number according to this:

**In the natural numbers, N, find the lowest such that half of it is a perfect square, one-third is a perfect cube, and one fifth is a perfect fifth power.**

Re: Need to find a number according to this:

Awww... no need to shout!

Sorry for misunderstanding - I already rectified it with a new response. :o

Re: Need to find a number according to this:

Re: Need to find a number according to this:

So the answer is 30,233,088,000,000? Wow! Thank you very much (Wink)