# Need to find a number according to this:

• Feb 23rd 2013, 02:11 AM
Sott
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
• Feb 23rd 2013, 05:41 AM
ILikeSerena
Re: Need to find a number according to this:
Quote:

Originally Posted by Sott
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?...n%20\mathbb{N}] with these features:

1) http://latex.codecogs.com/gif.latex?...20\frac{1}{2}x is a square of whole number

2) http://latex.codecogs.com/gif.latex?...20\frac{1}{3}x is a cube of whole nuber

3) http://latex.codecogs.com/gif.latex?...20\frac{1}{5}x is a fifth degree of whole number

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).
And x=3, x=4, x=5?
• Feb 23rd 2013, 06:33 AM
Sott
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...
• Feb 23rd 2013, 06:42 AM
Sott
Re: Need to find a number according to this:
• Feb 23rd 2013, 06:50 AM
ILikeSerena
Re: Need to find a number according to this:
(deleted)
• Feb 23rd 2013, 06:56 AM
ILikeSerena
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
• Feb 23rd 2013, 06:57 AM
Sott
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.

• Feb 23rd 2013, 07:01 AM
ILikeSerena
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
• Feb 23rd 2013, 07:04 AM
Sott
Re: Need to find a number according to this:
I am not shouting ;)
• Feb 23rd 2013, 07:07 AM
Sott
Re: Need to find a number according to this:
So the answer is 30,233,088,000,000? Wow! Thank you very much (Wink)