12. Distribute 572 peaches to several children, the number of peaches these children receive are some consecutive positive integers, then how many peaches can the child who gets the most peaches receive at most?
【Answer】 75
Analysis: Let the first person get $\mathrm{x}+1$ peaches, and the last person get $\mathrm{x}+\mathrm{k}$, then there are $\mathrm{k}$ people.
$$
572=(x+1+x+k) \times k \div 2=(2 x+k+1) k \div 2
$$
$1144=(2 \mathrm{x}+k+1) k, \mathrm{k}$ is a factor of 1144, $1144=11 \times 13 \times 2^{3}$
To find the maximum number of peaches the child who gets the most peaches can get, we need to minimize the number of people, which means $\mathrm{k}$ should be as small as possible. Enumerate $\mathrm{k}$ from small to large, $\mathrm{k}=2,4$ do not meet the requirements.
$k=8,2 x+9=143, x=67, x+k=75$, the child who gets the most peaches can get up to 75 peaches.