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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