it was narrated on the authority of abu ayyoob al-ansaari  that the prophet  said: "whoever fasts ramadhaan and then fasts six days of shawwaal, it is as if he fasts the whole year." [muslim]

it was narrated on the authority of thawbaan  that the prophet  said: "fasting ramadhaan equals fasting ten months, and fasting six days [of shawwaal] equals fasting two months. so both equal fasting the whole year." in another narration, he said: "whoever fasts six days after breaking the fast [of ramadhaan], it will be as if he fasted the whole year. allaah the almighty says (what means): {whoever comes [on the day of judgment] with a good deed will have ten times the like thereof [to his credit].} [quran 6:160]" [ahmad, ad-daarimi, ibn maajah, and an-nasaa’i] [ibn khuzaymah and ibn hibaan: saheeh]

benefits and rulings:

first: the virtue of fasting six days of the month of shawwaal, and that whoever keeps fasts them regularly after the month of ramadhaan will be as if he has observed fasting all his life. this is a huge merit and a great deed.

second: the mercy of allaah the almighty towards his slaves and his giving them huge rewards for their small deeds.

third: it is recommended to fast the six days immediately in response to the command of competing in righteous deeds and lest the muslim misses them or something distracts himher from fasting them.

fourth: it is permissible to fast the six days at the beginning, in the middle or at the end of shawwaal, consecutively or interruptedly. all this is permissible, and whatever the muslim chooses is permissible and worthy of reward should allaah the almighty accept it from him. [al-mughni and sharh an-nawawi]

fifth: the muslim who missed some days in ramadhaan should make up for these days first and then fast the six days of shawwaal based on the apparent meaning of the hadeeth. the prophet  said: "whoever fasts ramadhaan…" which means fasting the whole month, and this does not apply to the muslim who missed some days of ramadhaan until he makes up for them. moreover, freeing oneself from obligation is given precedence to doing a recommended act.

sixth: allaah the all-wise made the obligatory acts of worship preceded and followed by voluntary ones, such as the confirmed sunnah prayer before and after obligatory prayers as well as the permissibility of fasting during sha‘baan and six days of shawwaal while the obligatory fasting of ramadhaan lies between them.

seventh: voluntary acts of worship compensate for the imperfection taking place in the obligatory acts of worship. the muslim who is competent for religious assignment would definitely commit something which decreases the reward of his fast or blemishes it, such as unnecessary talk, uncontrolled looking, and the like.