Thursday, 25 August 2016

number theory - Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors



Prove that among any 12 consecutive positive integers

there is at least one which is smaller than the sum of
its proper divisors. (The proper divisors of a positive
integer n are all positive integers other than 1 and n
which divide n. For example, the proper divisors of 14 are 2
and 7)


Answer



Hint: Among any 12 consecutive positive integers, there is one that is a multiple of 12.



Can you show that 12n is smaller than the sum of its divisors for any positive integer n?


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