Assignment1-Solutions.pdf - Introduction to Combinatorics Assigment 1 University of Toronto Scarborough Name Student ID Solutions 1 Problem 1 Show that

Assignment1-Solutions.pdf - Introduction to Combinatorics...

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Introduction to Combinatorics: Assigment 1 University of Toronto Scarborough Name: Student ID: Solutions 1
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Problem 1. Show that given n + 2 natural numbers there are always two whose sum or difference is divisible by 2 n + 1. Solution (use the space below only). Consider the following pairs n + 1 pigeonholes { 0 } , { 1 , 2 n } , { 2 , 2 n - 1 } , , · · · , { n, n + 1 } . These numbers give us all the remainders when dividing by 2 n + 1. By PP two our numbers will be in the same set. If these two numbers coincide then their difference is divisible by 2 n + 2. If they do not then their sum will be divisible by 2 n + 2. Problem 2. Prove that there is a natural number composed with the digits 0 and 5 and divisible by 2019. Solution (use the space on this page below only). Consider the infinitely many numbers of the form 5 , 55 , 555 , .... By PP, two of them will give the same remainder when divided by 2019 and hence their difference will be a multiple of 2019. But their difference consists of 5’s and 0’s only.
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