Submitted by Atanu Chaudhuri on Sat, 18/05/2024 - 12:04
How to find all two digit consecutive integer sets with sum 100
Find all two digit sets of two or more consecutive positive integers that can be added to obtain a sum of 100. Follow a systematic approach.
The number puzzle is not simple.
Exhaustive approach to find all two digit consecutive integer sets with sum 100
To find all feasible number sets a systematic method is needed. Sequentially consider,
- Two 2 integer sets.
- One 2 integer and one 3 integer sets.
- Two 3 integer sets.
- One four integer set one 2 integer sets.
- One four integer set one three integer sets.
- Two four integer sets.
- One five integer set one 2 integer set
- One five integer set one three integer set.
- Three 2 integer sets.
Constraints
- Integer sets cannot have a number more than 40.
- None of the integers can be of three digits.
Two numbers of 2 integer sets: Systematic method starting from largest pair
- Largest pair (39,40) sum 79. Feasible pair (10,11).
- Reduce the largest number 40 by 1 up to (30,31) sum 61 with feasible pair (19,20). 10 such feasible pairs.
- Reducing more, next smaller feasible pairs: (29,30), (20,21); (28,29), (21,22); (27,28), (22,23); (26,27), (23,24); (25,26), (24,25). 5 such. Cumulative 15 feasible pairs.
One 2 integer and one 3 integer sets: Odd even property
- Sum of a 2 consecutive integer set is odd, whereas sum of 3 consecutive integer set will alternate between odd and even. Sum of the two must be even.
- 3 integer set largest feasible number will be (24,25,26) sum 75. Feasible pair (12,13).
- Smallest 3 integer set: (10,11,12) sum 33. Feasible pair (33,34).
- This range will have a total of 8 three integer sets starting with even integer with odd sum and a feasible two integer pair.
- Cumulative 23 feasible pairs.
Two three integer sets
- Sum of each set is divisible by 3, so is their sum. So no feasible two three integer set.
One 4 integer set, one 2 integer set
- Sum of 4 integer set will be even and of 2 integer set odd. No feasible pair.
One 4 integer set one three integer set
- Difference between 100 and sum of 4 integer set will have to be divisible by 6 to have a feasible three integer pair. Because, sum of three integer pair must be even and inherently divisible by 3.
- Only two such pairs are feasible: (10,11,12,13), (17,18,19) and (13, 14,15,16), (13,14,15).
- Cumulative feasible pairs 25.
Two four integer sets
- One feasible pair (10,11,12,13) and (12,13,14,15). Both should be larger than 40 and the first is the smallest set.
- Cumulative feasible pairs 26.
One five integer set, one 2 integer set
- Five integer sets are divisible by 5, less than 80 and must be odd to have a feasible two integer pair.
- The two such combinations are (11,12,13,14,15), (17,18) and (13,14,15,16,17), (12,13).
- Cumulative feasible pairs 28.
One five integer set and one three integer set
- The difference between 100 and sum of five integer set must be divisible by 5 and 3, that is 15, to have a feasible 3 integer pair. There is no such.
Three two integer sets
- Not feasible as sum of each will be odd.
Four two integer sets
- 10,11 = 21
- 11,12 = 23
- 12,13 = 25
- 13,14 = 27
- 14,15 = 29
- Only feasible quadruplet is (10,11), (11,12), (13,14) and (14,15).
Grand total of feasible pairs is 29. These are all.
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