"Which of the following must be an integer"-question

[rnbp9]

If "n" and "y" are positive integers and 450y=n^3, which of the following must be an integer?

I. y/(3)(2^2)(5)
II. y/(3^2)(2)(5)
III. y/(3)(2)(5^2)

a) none
b) I only
c) II only
d) III only
e) I, II, and III

First of all: 450 = 2*3^2*5^2
2*3^2*5^2*y=n^3

First I thought that I just need to compare this with the given solutions but it didnt work out because you cant choose "II and III"

Therefore, I did a lot of trial and error and got this:
450*60 = 30^3 (of course I wont have the time to do so in the actual test)
y=60 --> leaves just one option left

IMO B
Any approaches how to calculate his more efficient and less time-consuming?

[Abhishek009]

Therefore, I did a lot of trial and error and got this:

450*60 = 30^3

(of course I wont have the time to do so in the actual test)
y=60 --> leaves just one option left

Any approaches how to calculate his more efficient and less time-consuming?

Buddy it may be a trial and error method but U can learn something very important from it....

Lets see ;

450 * y

y can take any value 1,2,3,4,5,.............infinity

But when y multiplies with 450 the result will definitely lead the last digit as zero , let's check:

450*1 = 450

450*2 = 900

450*3 = 1350

450*6 = 2500

So on and so forth...

So it will be a tedious process....

Lets see the right hand part ; n^3

Now if the last digit of n^3 is 0 then n's last digit will also be zero !

Let's check 2^3 = 8

3^3 = 27

etc....

Now your work is reduced , to finding the last digits of n as zero...

Now go for it :

Case I

n = 10

n^3 = 1000

450*y = 1000

Hence y = 2.22 { Not Possible}

Case II

n = 20

450 *y = 20^3

450*y = 8000

y = 17.78 {Not possible}

Case III

n = 30

n^3 = 27000

450 * y = 27000

Hence y = 60 { Possible because both n and Y are positive integers }

I hope I am clear ; still if you need more explanation I will help u again...

[rnbp9]

Cool, thank you for the explanation.
It is totally clear that way!

It looks so easy but I coulnt get the right approach!

[Masuraha]

given: 450y=n^3
= 5^2*3^2*2*y=n^3
so minimum value of y=5*3*4

so answer is (1)