I have there two problems related with number theory and “peculiar descomposition …” I need a help with it… PROBLEM 1 Let’s divide all natural numbers except 1 into n non-empty sets so that: 1) each natural number except 1 is element of exactly one of these n sets 2) if two different commensurable numbers a, b are elements of the same set then also their greatest common divisor is element of this set. Prove that one of these n sets certainly contains all prime numbers besides maximally (n-1) exceptions (that signifies that “nearly” all prime numbers are in the same set – there can be maximally (n-1) prime numbers that may be in other sets). PROBLEM 2 Let’s divide all prime numbers into two infinite sets A,B so that each prime number is element of exactly one of sets A,B. Now let each composite number that is product of some primes only from A be element of A too and also let each composite number that is product of some primes only from B is element of B too. Let’s denote the set of the remaining composite numbers (such that they aren’t elements of A or B) as C. Prove that for each natural number n at least one of sets A,B or C contains n consecutive natural numbers. Could anybody give me some hints…? Thanks for all your replies…_________________

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