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This paper is concerned with the stability issue of discrete-time periodic positive systems with constant delays. First, a necessary and sufficient condition is proposed which tests whether or not a periodic system is positive. Then, another necessary and sufficient stability criterion is established which tests the asymptotic stability of periodic positive system with delays. It is well-known that for a linear time-invariant discrete-time positive system with or without delays, all the system matrices are nonnegative are necessary and sufficient to guarantee its positivity, and its asymptotic stability is not affected by the delays. However, this paper shows that, in the context of discrete-time periodic positive systems with delays, the above-mentioned features disappear.