What is the smallest positive integer n such that the fourth root of 56*n*360 is an integer?

Guest Apr 27, 2023

#1**+1 **

*What is the smallest positive integer n such that the fourth root of 56*n*360 is an integer?*

I don't know if it's the smallest integer, but I can help with __an__ answer. Here's what I did:

I broke down 56 and 360 into their smallest factors

36 = 2, 2, 2, and 7 and 360 = 2, 2, 2, 3, 3, and 5

Combined those and so 56 * 360 = 2, 2, 2, 7, 2, 2, 2, 3, 3, and 5

We want a perfect 4^{th} power, so I added enough of each factor to make 4 of them

The ones I added are

highlighted in dark blue 2, 2, 2, **2**, 7, **7**, **7**, **7**, 2, 2, 2, **2**, 3, 3, **3**, **3**, 5, **5**, **5**, and **5**

Multiply the added (blue) ones together 2 * 7 * 7 * 7 * 2 * 3 * 3 * 5 * 5 * 5 = 1,543,500

So **1,543,500** is one integer that will work. I just don't know if it's the smallest.

Check answer: 56 * 360 * 1,543,500 = 31,116,960,000 the 4^{th} root of which is 420

_{.}

Guest Apr 27, 2023