What is the ruling regarding the prayers of a person who performed ghusl without being in a state of janabah, without wudu, but included rinsing the mouth and nose and washing the entire body with water, believing that this ghusl sufficed for wudu, and does not know the number of prayers performed in this manner?

Ablution (ghusl) for cleanliness or cooling down is not a substitute for ritual ablution (wudu'). Therefore, the prayers performed after such a ghusl without wudu' are invalid, and they must be repeated due to the absence of the condition of purity (tahara). If the number of invalid prayers cannot be determined, then one must continue to make up for them until it is highly probable that one's responsibility has been cleared. The preponderant opinion, which is that of the majority of scholars, is that the prayer of one who omits a condition or a pillar, whether out of ignorance or forgetfulness, is invalid.