1997 AHSME Problems/Problem 20
Contents
Problem
Which one of the following integers can be expressed as the sum of consecutive positive integers?
Solution
The sum of the first integers is .
If you add an integer to each of the numbers, you get , which is the sum of the numbers from to .
You're only adding multiples of , so the last two digits will remain unchanged.
Thus, the only possible answer is , because the last two digits are .
As an aside, if , then , and the numbers added are the integers from to .
Solution
Notice how the sum of 100 consecutive integers is .
Cancelling out the constants give us .
Looking over at the list of possible values, we quickly realise that the only possible solution is
See also
1997 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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