Suppose 14 students in a class appear at an examination. Prove that there exist atleast two among them whose seat numbers differ by a multiple of 13.
Answers
Answer:
14 वुद्यक्फ़जयच्यवौइज्येट्यू I0I
Answer:
PROOF:
Since there are 14 different numbers ∈ N> 13, apply the division algorithm to get the remainder 0-12 of those numbers 13. Therefore, create a set {13q0 + r0,13q1 + r1, ..., 13q12 + r12}. Residues r1 to r12 are numbers from 0 to 12 and are different. If they are all the same, the difference is divisible by 13. That's all. Here, the 14th number must have the same remainder as one of the other 13, so the difference is divisible by 13. I actually found this in the guidance proof. Derivation of number range / maximum number n.
Basic case n = 14: All numbers from 1 to 14 are required. 14-1 = 13. Suppose it is true for
n. For n + 1, the above applies. If all 14 are selected from the set of ranges n, this is executed / true by the inductive hypothesis.
Otherwise, n + 1 can select a set / subset {{i1, i2, ..., i13}, n + 1} and apply the above. That is, if none of {i1, i2, ..., i13} have the same mod 13 residues, then they cover the entire range of mod 13 residues, so n + 1 mod 13 is {i1, i2. ,. Must be equal to one of. .., i13} mod13. This completes the guidance. (If the difference between the two mods 13 is divisible by 13)
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