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 2012-02-19, 17:02 #1 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 3·419 Posts 2/3 Powers being viewed over the Ring Z/(10^n)Z 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 $\phi(n)$, 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 $2^{20} = 76 (mod 100)$ => is being the identity element over Z/100Z $2^{100} = 376 (mod 1000)$ => is being the identity element over Z/1000Z $2^{500} = 9376 (mod 10000)$ $2^{2500} = 9376 (mod 100000)$ $2^{12500} = 109376 (mod 1000000)$ $2^{62500} = 7109376 (mod 10000000)$ ... Consider this, rather $3^{20} = 1 (mod 100)$ $3^{100} = 1 (mod 1000)$ $3^{500} = 1 (mod 10000)$ $3^{2500} \ne 1 (mod 100000)$ => is Not being the identity element at all $3^{12500} \ne 1 (mod 1000000)$ => WHY? $3^{62500} \ne 1 (mod 10000000)$