Is Any Carmichael Number Divisible by a Perfect Square Greater Than 1?

[CornMuffin]

Homework Statement

Prove or disprove (and salvage if possible):
No Carmichael Number is divisible by a perfect square > 1

Homework Equations

A composite number n is called a Carmichael number if and only if $$a^{n-1} \equiv 1 (mod \ n)$$ for all $$2\leq a \leq n-1$$ such that gcd(a,n) = 1

Carmichael numbers comes from fermat's little theorem that states that all prime numbers have this property. Carmichael numbers are numbers that have this property but are composite.

The Attempt at a Solution

I have been trying to figure out a way to do this problem for awhile, but no luck

 

[Office_Shredder]

What if there exists a prime p such that p² divides a?

You might want to draw some inspiration from the simplest case (what if a=p²)