This is what we found on Wikipedia regarding the Ten Hail Marys:

Three Hail Marys are a traditional Roman Catholic devotional practice of reciting Hail Marys as a petition for purity and other virtues. Believers recommend that it be prayed after waking in the morning, and before going to bed. This devotion has been recommended by SS. Anthony of Padua, Alphonsus Liguori, John Bosco and Leonard of Port Maurice. Two saints, Mechtilde and Gertrude the Great, are said to have received revelations from the Blessed Virgin Mary regarding this practice.

It is a common practice for Catholics to offer three Hail Marys for any given problem or petition.

Pope Leo XIII granted an indulgence to those who practice the Three Hail Marys devotion and Pope Benedict XV raised the Confraternity of Three Hail Marys to the Archconfraternity of Three Hail Marys.