Suppose they meet today, as the 1 day,

A Go to 1 time every week, that is, on the eighth day, 15, 22, 29, 36, 43, 50. ...

B I went 1 time in three days, that is, I went on the 4th and 7th, 10, 13, 16, 22, 25, 28, ...

It can be inferred from the above that Party A and Party B will meet on 22- 1 day, that is, 2 1 day.

Calculation method:

Go once a 7 days and once B 3 days. The least common multiple of 3 and 7 is 2 1, so in 2 1 days, A and B will meet in the library.