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If you sum a sequence with period $a$ and a sequence with period $b$, then you get a sequence which is $\mathsf{LCM}(a,b)$-periodic. But the new sequence might have a smaller period, as the following example demonstrates (addition is modulo 10): 022441133502244113350224411335 000772999611888000772999611888 022113022113022113022113022113 The first sequence ...


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A Vigenère cypher on an alphabet of size $M$, which we identify with the integers modulo $M$, works as follows. The key is an arbitrary word, $k_0 \ldots k_{\ell-1}$. Given a plaintext $p_0 \ldots p_{n-1}$, the ciphertext $c_0 \ldots c_{n-1}$ is given by $$ c_i = p_i + k_{i \bmod \ell} \bmod M. $$ For example, suppose that we are trying to encipher decimal ...


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