Shawwal is the tenth month in the lunar calendar. The first of Shawwal is Eid ul Fitr. After the festivity of Eid it is recommended to observe six days of fast. This fast may be observed continuously non-break, or it may be observed one day at a time. If you observe it continuously, you may start on the fourth day and end on the ninth of day Shawwal, or you may select days at random, provided you complete six days before the end of Shawwal. For instance, you may observe the 3rd, 5th, 7th, 9th, 14th and 15th days. Abu Ayyub Al-Ansari (May Allah be pleased with him) related the Messenger of Allah, (p.b.u.h.), said: "Whoever observes the Ramadan fast and follows it with six days of fast in Shawwal, it is as if he has fasted Dahr (the whole year or possibly forever)." (Bukhari)

Analyzing this Hadith, our jurists (Ulama) explained how according to this Hadith, a Muslim who fasts during Ramadan every year and follows it with six days fast of Shawwal, will be credited for fasting a whole lifetime. The Jurists correctly said: "A good deed (Hasanah) is rewarded a minimum of ten times its equivalent. It follows, then, that one Ramadan is equivalent to ten months of fasting, and the clincher, six days, is equal to two months, (6x10=60)." That undoubtedly completes the year's twelve months. Thus, we see the wisdom and the reason why the Prophet (p.b.u.h.) mentioned six days after Ramadan in Shawwal, not five or seven.

Source: Jannah.org