1. Does there exist 100 positive integers such that their sum equals their least common multiple?

143
［Solution］Exists.
In fact, we can take the following 102 numbers:
$$1,2,2^{2}, 2^{3}, \cdots, 2^{67}$$

and $3,3 \cdot 2,3 \cdot 2^{3}, 3 \cdot 2^{5}, \cdots, 3 \cdot 2^{65}$.
Their sum is
$$2^{68}-1+3+3 \cdot 2 \cdot \frac{\left(2^{2}\right)^{33}-1}{2^{2}-1}=3 \cdot 2^{67}$$

Their least common multiple is also $3 \cdot 2^{67}$.
By replacing the numbers 4 and 8 with 12, 16, and 32 with 48 in these 102 numbers, we obtain 100 numbers. Clearly, the sum and the least common multiple of these 100 numbers are both $3 \cdot 2^{67}$.