SOLUTION: Express 1994 as a sum of coonsecutive positive integers and show this is the only way to do it.

Question 42055: Express 1994 as a sum of coonsecutive positive integers and show this is the only way to do it.
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi,
1994 can be written as 497+498+499+500 so there is how it's done. Proving this is the only way is quite simple but it's quite long and I don't really want to type it here. An outline of the proof is as follows.
1) To write a number N as the sum of (possibly negative) consecutive integers, where the first term is a. Then

2) We can rearrage to find a

3) We want a to be an integer so (replace equals with congruent to)

4) (Be careful with 2N congruent 0 mod 2n here!) If n is odd then this has a solution when n divides N. If n is even then there is solution when n does not divide N, but does divide 2N.
5) So taking N=1994, the only values of n that satisfy the above conditions are n=4,997,3988. (N=1 is obviously a trivial solution, but you really want n>1 right?)
6) Calculating a for the above 3 values gives 497, -496, and -1993. You are only interested in the sums which use strictly posotive numbers. There is only one of those, with n=4 which I gave you above.
Hope that helps,
Kev
RELATED QUESTIONS
Show that there is only one set of different positive integers, x,y,z that 1=... (answered by Edwin McCravy)

The sum of two consecutive positive integers is at most 7. Find the integers. My teacher... (answered by Alan3354)

What is the smallest positive integer that can be expressed as the sum of nine... (answered by Fombitz)

It is possible to write 18 as the sum of two or more consecutive positive digits in two... (answered by ikleyn)

What are the last two digits of: (A)3 to the power of 1994 (B)7 to the power of 1994... (answered by stanbon)

PROBLEM: The integer 48 is to be expressed as a sum of x consecutive odd integers. The (answered by sudhanshu_kmr)

1. what is the smallest integers that can be express as the sum of the square of the two... (answered by richard1234)

the integer 33 is to be expressed as a sum of n consecutive positive integers. The value... (answered by Edwin McCravy)

I have been trying this problem and I am stuck and need help! my final is monday. One... (answered by Alan3354,stanbon)