CONGRUENT: Let us start by choosing a base n, then each integer = q.n + r where q is quitent and r is remainder (0< r<n). We write this as Ner (mod n) and say that N is congruent to r modulo n. e.g. 78 =66 (mod 12).
Ex. January 1, 2000 falls on a saturday. What day of the week will January 1, 2020 be?
Sol. Because there are 20 years in the range 2000 to 2019 of which 5 are leap-years, January 1, 2020 falls on day 20 + 5 = 4(mod 7), i.e. Wednesday.
MATHEMATICAL INDUCTION :
(1) Weak Version: Let P(n) be a statement such that (i) P (n0) is true and (ii) P(k) implies P(k+l) for any k>n0, then P(n) is true for every n >n0.
(2) Strong Version : Let P(n) be a statement such that (1) P(n0) is true and (ii) if P(n0), P(n0+1),......,P(k) are true for any k > n0, then P(k+1) is also true. Then P(n) is true for every n>n0.
PIGEONHOLE PRINCIPLE : If m pigeons are assigned to n pigeonholes , where m>n, then at least two pigeons must occupy the same pigeonhole.
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