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MMO 276 Moscow MO 1954 divisors of product of primes 2 to 31, combo sum

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August 13, 2019
combinatoricsprimesProductSum

Problem Statement

a) Let 1,2,3,5,6,7,10,..,N1, 2, 3, 5, 6, 7, 10, .., N be all the divisors of N=2⋅3⋅5⋅7⋅11⋅13⋅17⋅19⋅23⋅29⋅31N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 (the product of primes 22 to 3131) written in increasing order. Below this series of divisors, write the following series of 11’s or −1-1’s: write 11 below any number that factors into an even number of prime factors and below a 11, write −1-1 below the remaining numbers. Prove that the sum of the series of 11’s and −1-1’s is equal to 00.
b) Let 1,2,3,5,6,7,10,..,N1, 2, 3, 5, 6, 7, 10, .., N be all the divisors of N=2⋅3⋅5⋅7⋅11⋅13⋅17⋅19⋅23⋅29⋅31⋅37N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 (the product of primes 22 to 3737) written in increasing order. Below this series of divisors, write the following series of 11’s or −1-1’s: write 11 below any number that factors into an even number of prime factors and below a 11, write −1-1 below the remaining numbers. Prove that the sum of the series of 11’s and −1-1’s is equal to 00.
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