How many two digit consecutive integer sets have sum 100?

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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