Substitute each of the small letters in bold with a different decimal digit from 0 to 9, to satisfy the following system of simultaneous modular alphametic equations:

one ≡ 1 (mod two), and:
one ≡ 1 (mod six), and:
six ≡ 0 (mod two)

where, none of s, o and t is zero.

Note: Try to derive a non computer-assisted solution, although computer program/spreadsheet solutions are welcome.