2006 AMC 10A Problems/Problem 9

Problem

How many sets of two or more consecutive positive integers have a sum of 15?

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$

Solution

At a first glance, you should see that 7+8=15.

But are there three consecutive integers that add up to 15? Solve the equation

$n+n+1+n+2=15$ , and you come up with n=4. 4+5+6=15.

Again solve the similar equation

$n+n+1+n+2+n+3=15$ to determine if there are any four consecutive integers that add up to 15. This comes out with the non-integral solution 9/4. Further speculation shows that 1+2+3+4+5 = 15. So the answer is (C). 3

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2006 AMC 10A Problems/Problem 9

Problem

Solution

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