The number 21 is the sum of two consecutive integers 10 and 11. What is the largest number of positive, consecutive integers whose sum is 21?
First, we see that $1+2+3+4+5+6=21$.  If a number is the sum of seven or more consecutive positive integers, then the number must be at least $1 + 2 + \dots + 7 = 7 \cdot 8/2 = 28$, so $\boxed{6}$ is the largest number of consecutive integers we can use.