- #1

- 149

- 0

## Homework Statement

Let N be the smallest integer such that 10 x N is a perfect square and 6 x N is a perfect cube.

How many positive factors does N have?

(options)

A 30 B 40 C 54 D 72 E 96

## Homework Equations

## The Attempt at a Solution

right, Ill talk you through my attempt so far...

so,

I first of all tried a few values for N (the basics as tests like 0,1,2,3,5,10, etc..) and came to the logical conclusion that it wasn't going to be as simple as that, and probably would be a very large number (probably should have concluded that from the options but o well

o yeh, I found out that 0 kinda works, but I'm guessing from the options that 0 doesn't count because 0 isn't a cube or a square? am I right?

Then I tried using modular arithematic (I still a super noobie with it)

N = 0 (mod 10) is a perfect square, while N = 0 (mod 6) is a perfect cube.

From that I started working my way through the first half, trying lots of options for what makes the first bit work. I came up with stuff like N = 10, 40, 90, etc.. Not in a very logical approach but hey, mighta saved me time, but it didnt...

I can't figure out how to find N?

Have I approached the question all wrong though? is finding N entirely necessary or can you derive the factors somehow?

I tried the later approach, looking for similarities between the factors of squares and cubes but I don't know if it was me being slow, or wierd, or whatever, (or oblvivious to the obvious) but I couldn't find anything that way...

any help please :D

and as per usualy, one step at a time ;)

thnx