I am doing very severe load coursework for Ph.D. from IMSc, Chennai,

that's why I am not able tending to be active over this forum for the past year at all.

Very recently, I had come across some very fascinating property, for this,

I'd like (need) to seek the answer for this.

Euler's Theorem conveys necessarily that the order for an element over (mod n) is always being a divisor for

,

but though the properties for the powers of 2 & 3 vary accordingly as follows.

Why is it being so, the properties are rather being different from them apart?

As follows

=> is being the identity element over Z/100Z

=> is being the identity element over Z/1000Z

...

Consider this, rather

=> is Not being the identity element at all

=> WHY?