How many positive integers from 1 to 100 inclusive can be written as the sum of two or more consecutive positive integers? (Different examples 10 and 51: 10 = 1 + 2 + 3 + 4 and 51 = 25 + 26)

Source: mathcontest.olemiss.edu 3/7/2011

**SOLUTION**

**Sum of two consecutive numbers**

There are 100 numbers from 1 to 100. Half of them are even and half are odd. All the above sums are odd. Thus, there are 49 numbers from 3 to 99 that can be written as the sum of two consecutive positive integers.

**Sum of three consecutive integers**

duplicates are indicated by underlines.

An easy way to count these sums is to think of them as multiples of 6:

Thus, sixteen numbers {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96} can be written as the sum of three consecutive positive integers.

**Sum of four consecutive integers**

Sixteen numbers {10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98} can be written as the sum four consecutive positive integers.

**Sum of five consecutive integers**

Surprise! 30 is a duplicate but 20 is not.

Four numbers {20, 40, 80, 100} can be written as the sum of five consecutive positive integers.

**Sum of six consecutive integers**

Luckily, all the sums are odd and therefore duplicates.

**Sum of seven consecutive numbers**

Two numbers {28, 56} can be written as the sum of seven consecutive positive integers.

**Sum of eight consecutive integers**

Five numbers {44, 52, 68, 76, 92} can be written as the sum of eight consecutive positive integers.

**Sum of nine consecutive integers**

There are seven numbers {45, 54, 63, 72, 81, 90, 99} in this category but they are duplicates.

**Sum of ten consecutive integers**

There are six numbers {55, 65, 75, 85, 95} in this category but they are duplicates.

**Sum of eleven consecutive integers**

There are four numbers {66, 77, 88, 99} in this category; all are duplicates except 88. Thus, there is one number {88} that can be written as the sum of eleven consecutive positive integers.

**Sum of twelve consecutive integers**

There are two numbers {78, 90} in this category; both are duplicates.

**Sum of thirteen consecutive integers**

There is one number {91} in this category; it is a duplicate.

**Sum of fourteen consecutive integers**

.

The sum is greater than 100. So, we stop here.

**Summary**

Sum of two consecutive integers: 49 numbers

Sum of three consecutive integers: 16

Sum of four consecutive integers: 16

Sum of five consecutive integers: 4

Sum of six consecutive integers: 0

Sum of seven consecutive integers: 2

Sum of eight consecutive integers: 5

Sum of nine consecutive integers: 0

Sum of ten consecutive integers: 0

Sum of eleven consecutive integers: 1

Sum of twelve consecutive integers: 0

Sum of thirteen consecutive integers: 0

Total =

**Answer**: 93.

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