Ninety-nine apples are distributed among some boys in such a way that all receive different quantities of apples.

(a) What is the maximum number of boys that can be in this group?

(b) If there are ten boys, what is the maximum number of apples that the boy who received the fewest apples can receive?

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(a) To maximize the number of boys, we need to minimize the number of apples each one can receive. In this case, the first natural numbers $1,2,3,4, \ldots$, correspond to the quantities of apples each boy should receive, except for the last boy. Since

$$
1+2+3+\cdots+12+13=91
$$

and

$$
1+2+3+\cdots+13+14=105
$$

the maximum number of boys is 13.

(b) Let's observe that

$$
1+2+\cdots+10=55
$$

is the minimum number of apples that the ten boys receive. For each apple we give to the boy with the fewest apples, we must give an apple to each of the others so that everyone ends up with distinct quantities of apples.

Since $99-55=44$, we can give 4 additional apples to each of the boys. Therefore, the boy with the fewest apples can receive a maximum of 5 apples (Note that $5+6+\cdots+14=95$ and $6+7+\cdots+15=105$).