Thread: Need to find a number according to this:

1. 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 [ $\dpi{200} x \epsilon \mathbb{N}$] with these features:

1) $\dpi{120} \frac{1}{2}x$ is a square of whole number

2) $\dpi{120} \frac{1}{3}x$ is a cube of whole nuber

3) $\dpi{120} \frac{1}{5}x$ 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

2. Re: Need to find a number according to this:

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 [ $\dpi{200} x \epsilon \mathbb{N}$] with these features:

1) $\dpi{120} \frac{1}{2}x$ is a square of whole number

2) $\dpi{120} \frac{1}{3}x$ is a cube of whole nuber

3) $\dpi{120} \frac{1}{5}x$ 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 $\mathbb N$ starts with 1).
What about x=2?
And x=3, x=4, x=5?

3. 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...

4. Re: Need to find a number according to this:

$\dpi{150} \frac{1}{2}x = a^{2}$

$\dpi{150} \frac{1}{3}x = b^{3}$

$\dpi{150} \frac{1}{5}x = c^{5}$

$\dpi{150} x \epsilon \mathbb{N}$ and $\dpi{150} (a; b; c) \epsilon \mathbb{Z}$

(deleted)

6. Re: Need to find a number according to this:

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

7. 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.

8. 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.

9. Re: Need to find a number according to this:

I am not shouting

10. Re: Need to find a number according to this:

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