Two consecutive integers -- 105 divided by 2 = 52, 52 + 1 = 53
52 and 53
Three consecutive integers -- 105 divided by 3 = 35, 35 - 1 = 34, 35 + 1 = 36
34, 35 and 36.
Four consecutive integers -- can't be done, since the sum of any four consecutive integers is an even number.
Five consecutive integers -- 105 divided by 5 = 21, 21 - 2 = 19, 21 - 1 = 20, 21 + 1 = 22, 21 + 2 = 23
19, 20, 21, 22, 23
Six consecutive integers -- 105 divided by 6 = 17, and 105 divided by 3 = 35, so use the consecutive integers that add up to 35, namely 17 and 18, 16 and 19, and 15 and 20.
15, 16, 17, 18, 19, and 20.
Seven consecutive integers -- 105 divided by 7 = 15, by the same process shown above for 3 and 5,
12, 13, 14, 15, 16, 17, and 18.
Eight consecutive integers -- Can't be done, since the sum of any eight consecutive integers is an even number.
9 consecutive integers -- 105 divided by 9 = 11, remainder 6, so using 11 as the midpoint of the nine numbers would result in 7 and 15, 8 and 14, 9 and 13, 10 and 12, and 11, for a total of 99. Shifting the sequence up one number would result in a total of 108, so it can't be done.
10 consecutive numbers -- 105 divided by 10 = 10, remainder 5, so using 10 and 11 as the midpoint, you can use 6 and 15, 7 and 14, 8 and 13, 9 and 12, and 10 and 11.
6,7,8,9,10,11,12,13,14,and 15
11 consecutive numbers -- 105 divided by 11 = 9, remainder 6. Using a midpoint of 9, it would be 4 and 14, 5 and 13, 6 and 12, 7 and 11, 8 and 10, and 9. This totals 99. Shifting the sequence up one number would result in a total of 110, so it can't be done.
12 consecutive numbers -- Can't be done, since the sum of any 12 consecutive numbers is an even number.
13 consecutive numbers -- 105 divided by 13 is 8, with a remaindeer of 1. Can't be done (see 14 numbers)
14 consecutive numbers -- The sum of the first 14 consecutive integers is 105.
1,2,3,4,5,6,7,8,9,10, 11, 12, 13, and 14.
So it can be done with 2, 3, 5, 6, 7, 10 and 14 consecutive integers.
Thus 7 different ways can 105 be written as the sum of two or more consecutive positive integers. Ans
