Question 1080516: Hi, how would I work this question out? Thank for your help in advance:)
A partition of an integer into distinct positive parts (no two are equal) is called an unequal partition. For example, there are seven unequal partitions of 12 into 3 parts: 12= 9+2+1 = 8+3+1 = 7+4+1 = 7+3+2 = 6+5+1 = 6+4+2 = 5+4+3.
The span of a partition is the difference between the smallest and largest number in the partition. For example, the spans of the unequal partitions of 12 into 3 parts listed above are respectively 8,7,6,5,5,4,2. Thus 2 is the smallest span of an unequal partition of 12 into 3 parts.
a. Find at least two unequal partitions of 2017 into 5 parts with a span of 7.
b. Find an unequal partition of 2017 into 5 parts with the smallest possible span.
c. Find the smallest number that has an unequal partition into 5 parts. Describe the set of all numbers that have an unequal partition into 5 parts with a span of 4.
d. Show that for each number that has an unequal partition into 5 parts, the smallest span for such partitions is 4 or 5.
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! I'll at least get you started.
Such a large number with a small span means the numbers are close to each other.
Dividing 2017 by 5 gives you 404.5.
So start with numbers around there.
I used EXCEL typing numbers into 4 cells and calculating the last one (since the sum equals 2017).
Then changed numbers around.
(401,402,403,404,407) gives you 6.
(401,402,403,405,406) gives you 5.
(399,403,404,405,406) gives you 7.
a) Play around to get another.
b) I think its 5 but play around to see if you can find one lower.
c) It should be,
 with a span of 4.
All numbers would then increment by 1,
So the sums would increase by 5(1) or 5.
15,20,25,30,35,40,.... with a span of 4.
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