Abstract
where the mod operator on the right is the usual mathematical one, with the convention that x mod 0 = x, so that the definition makes sense even when a = b. That is, following [4, p. 82], x mod y = x− y⌊x/y⌋ for y 6= 0, and x mod 0 = x. We use Definition (1) because we want the usual mathematical mod (when a = 0) as a special case. Definition (1) works perfectly well when the interval is given backward, that is, a > b, yielding a modulus in the right-closed interval (b . . a]. It follows from Definition (1) that a ≤ x mod [a . . b) b. Definition (1) allows us to define the analogous equivalence relation,
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