Question

#3 and 5 only

3. Prove that if six natural numbers are chosen at random, then the sum or difference of two of them is divisible by 9. 4. Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within 2 units of each other. 5. Prove that any set of seven distinct natural numbers contains a pair of numbers whose sum or difference is divisible by 10.

Answer 1

of to Prove that any of set of seven distinct natural numbers contains a plut of numbers whose Sum or difference is divisible by lo. det's partition the seven numbers into six categories lo Multiples 20 Thase with lor q in the ones digit 3. Those with or 8 in the ones digit Yo Those with 3 of 7 in the ones digit 5. Those with yor 6 in the ones digit. 6. Those with 5 in the ones digit By the pigeonhole principle, when 7 am numbers are categorized this way, there must be at least one Category among these containing a numbers. call those numbers xog case 1 If xd y have the same ones digit then xy is disisible by to. case 2 If xly have different ones digit. But the above categories were selected so that each category has at most a single pair each category which sum to do divisible by " alty is a mula divisible by 10.

- & Prove that it six natural numbers are chosen then the at random, a sum ar difference dissible by 9. of two of them is s Proof we will prove it using pigeon hole prin ciple! Pigeon hole principle! If we sort more than nobjects into n pigeon holes, at least one pigeon hole will contain at least two objects. det six numbers be a, b, c, d, e and to If two of to them are divisible by 9, then results follows. Let none of b, c, d, ent are divisible by 9. and a may or may not be divisible by 9.0 atx= a-y mod q aqrelian 3 mod 9. » xty so atxe aty mod q > x-y = o moda . We have xery oux-y is divisible by 9 If we take x & y belongs to {b, c, d, et Hence, result follows