Katedra matematiky - Katolícka univerzita v Ružomberku

66 Jan Kopka, Leonard Frobisher, George FeissnerProblem: Investigate the number of different ways a given positive integercan be written the sum of consecutive positive integers.Here is our investigation:Again we can start with the process of experimentation. We can also use ourmini-theory, which was created above. But first several specific questions.Question: Which numbers have no representation as the sum of consecutiveintegers?Answer: Powers of 2.Question: Which numbers have a unique representation?Answer: Those with only one odd factor other than 1, namely primes andproducts of a power of 2 and a prime, such as 56 = 8·7 = 2 3 ·7.Question: Which numbers have exactly two such representations?Answer: Those with exactly two distinct odd factors other than 1, such as9 = 3 2 . The two distinct factors are 3 and 9.Question: Which numbers have exactly three such representations?Answer: Those with exactly three distinct odd factors other than 1, suchas 42 = 2·3·7. The three distinct odd factors are 3,7, and 21.We are now in a position to offer a further conjecture.Conjecture 4: Let n be positive integer. The number of representationsof n as a consecutive sum is the same as the number of odd factors of theinteger n.We are pretty certain that the conjecture is true, but can we prove it?Proof: For every odd factor 2k+1 of n we can construct sum (2) and aftercounteract we always get a sum of consecutive positive integers.For example: Number 30 = 2.3.5 has three odd factors: 3, 5 and 3.5 = 15.Sum of 3 consecutive integers: 30 = 9+10+11Sum of 5 consecutive integers: 30 = 4+5+6+7+8Sum of 15 consecutive integers (or 4 consecutive positive integers):30 = (−5)+(−4)+(−3)+(−2)+(−1)+0+1+2+3+4+5+6+7+8+9=6+7+8+9.The conjecture 4 can now be claimed to be a theorem 5.We end with two remarks of a didactic nature:Remark 1: We can use a geometric approach to introduce younger studentsto this problem with the help of figures such as these: