# find the numberfind the natural number if (6n+15)/3n is natural

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The problem provides the information that (6n+15)/3n is natural, hence, 3n divides exactly 6n + 15.

(6n+15)/3n = 2 + 15/(3n) = > (6n+15)/3n = 2 + 5/n

Notice that 2 is a natural number, hence, 5/n also needs to be natural, thus, n divides exactly 5 => n = {1;5}.

Hence, substituting 1 for n yields:

(6+15)/3 = 21/3 =>(6+15)/3 = 7

Hence, substituting 5 for n yields:

(6*5+15)/15 = 45/15 => (6*5+15)/15 = 3

**Hence, evaluating the number under the given conditions yields (6n+15)/3n = 3 or (6n+15)/3n = 7.**

(6n+15)/3n = K where k is a natural number ( 1,2,3,4...)

Let us simplify.

==> 6n/3n + 15/3n = k

==> 2 + 5/n = k

We have 2 is a natural number.

Then 5/n must we a natural number too.

==> 5/n = K

==> n = 1 or 5

Let us substitute with n=1

==> (6n+15)/3n = (6+15)/3 = 21/3 = 7

Now we will substitute with n= 5

==> (6n+15)/3n = (6*5 + 15)/3*5 = 45/15 = 3

**Then, the number is 3 or 7.**

First, we'll factorize the numerator by 3:

3(2n + 5)/3n

We'll divide by 3:

(2n + 5)/n = (n + n + 5)/n = n/n + n/n + 5/n

We'll simplify and we'll get:

(2n + 5)/n = 1 + 1 + 5/n

(2n + 5)/n = 2 + 5/n

For the number (2n + 5)/n to be natural, n has to be a positive divisor of 5.

D5 = +1 and +5

For n =1

(2n + 5)/n = 2 + 5/1

(2n + 5)/n = 2 + 5 = 7

For n = 5

(2n + 5)/n = 2 + 5/5

(2n + 5)/n = 2 + 1 = 3

**The natural number could be 3 or 7.**