smallest value?

[infinityzero]

n and p are integers greater than 1

5n is the square of a number

75np is the cube of a number.

The smallest value for n + p is

A. 14

B. 18

C. 20

D. 30

E. 50

 

 

 

 

 

Ans:A

[cybernike]

5n= the square of a number

it is not difficult to see n=5.

 

75*n*p

=75*5*p

=3*5*5*5*p

=3*p*5^3=the cube of a number.

In order to get a cube, all the prime components (i.e. 3 and 5 here) have to be to the 3rd power. Therefore, the smallest number for p is 3^2 =9.

 

Therefore, n+p= 5 +9 =14 (A).

 

n and p are integers greater than 1

5n is the square of a number

75np is the cube of a number.

The smallest value for n + p is

A. 14

B. 18

C. 20

D. 30

E. 50

 

 

 

 

 

 

Ans:A

[infinityzero]

"5n= the square of a number

it is not difficult to see n=5."

 

How does n = 5???

 

I did 5n = x^2 and 75np = x^3, where x is "a number". I divided 75np by 5n to get 15p, and 15p = x:

 

(75np/5n) = (x^3/x^2) --> 15p = x

 

Since x represents a number (and not an actual variable in this problem), I said that 15 = p/x or 15 = p.

 

Then 75*(15)*n = 5*5*3*5*3*n

p = 9 and I would have to solve for n.

 

I don't understand how n=5 automatically here though. Can someone explain? Thanks.

[kronique]

"5n= the square of a number

it is not difficult to see n=5."

 

How does n = 5??

n is five here because we need to find the smallest vaule of n+p this is possible only when both n & p are min.

obviously for 5n to be a square we can have many values on n...namely 5, 20,80 etc.

but we are interested in the smallest value of n. Hence n=5.

 

I did 5n = x^2 and 75np = x^3, where x is "a number". I divided 75np by 5n to get 15p, and 15p = x:

 

(75np/5n) = (x^3/x^2) --> 15p = x

 

The highlighted part is wrong infinity, because by that statement,you are assuming that 5n is the square of x while 75np is the cube of the same no. x !!

(This need'nt be always the case.You are considering a very special case.)

Thinking more generally, as cybernike pointed out n=5.

Now we have to find out the smallest value of p.

as cybernike says , again..

 

75*n*p

=75*5*p

=3*5*5*5*p

=3*p*5^3=the cube of a number.

In order to get a cube, all the prime components (i.e. 3 and 5 here) have to be to the 3rd power. Therefore, the smallest number for p is 3^2 =9.

 

now from this, only if p=9,73, etc can the no. 3*p*(5^3) be the cube of any possible no.

 

Again the lowest allowed value of p as we see is 9. Hence n =5 & p=9. so the smallest value of n+p = 14.

 

Hope this is clear.

[infinityzero]

Yes, ok. I understand. I was writing things out in an equation format, hoping to solve for n or p. But it seems that is not the case for this problem and 5n and 75np must be taken by themselves alone and not combine together to solve for n or p, which is what I thought was to be done (and am familiar with).

 

Well, thank you both :)