The Messenger of Allah ﷺ said, "Whoever fasts three days of every month, that is equivalent to fasting the entire year." Then Allah, the Mighty and Majestic, revealed confirmation of this in His Book, "Whoever brings a good deed shall have ten times the like of it" [Anam 6:160]. So one day counts as ten days.
قال رسول الله ﷺ: من صام من كل شهر ثلاثة أيام، فذلك صيام الدهر. فأنزل الله عزه وجل تصديق ذلك في كتابه {من جاء بالحسنة فله عشر أمثالها} [سورة الأنعام: ١٦٠] فاليوم بعشرة أيام.
It was narrated by Tirmidhi (762), Nasai (2409), Ibn Majah (1708), and Ahmad (21301), all of them through various routes from Asim Ahwal, from Abu Uthman, from Abu Dharr, as mentioned. Tirmidhi said, "A hasan sahih hadith. Shubah narrated this hadith from Abu Umar and Abu Tayah, from Abu Uthman, from Abu Hurairah, from the Prophet ﷺ."Ibn Khuzayma (3/302) alluded to the report of Abu Uthman from Abu Dharr, and perhaps he included it in his larger book. Munziri mentioned it in Targheeb wa Tarheeb (1579) and attributed it to those I have attributed it to, and he affirmed Tirmidhi's grading of it as hasan and Ibn Khuzayma's grading of it as sahih.Abu Uthman is Abdul-Rahman bin Mull bin Amr Nahdi Kufi. He witnessed the pre-Islamic era, embraced Islam during the time of the Prophet ﷺ, but never met him. Abu Hasan bin Bara said, "I copied this from the book of Ali bin Madini, though I did not hear it from him directly, 'Abu Uthman Nahdi, whose name was Abdul-Rahman bin Mull, also said to be Mull. He was originally from Kufa and later moved to Basra. He was of Arab stock and witnessed the pre-Islamic era. He migrated to Madinah after the death of Abu Bakr, coinciding with the succession of Umar. He heard from Umar but did not hear from Abu Dharr.'"This statement, "he did not hear from Abu Dharr," is questionable, because Abu Dharr was in Madinah at the time of Umar's succession, and he did not leave for Rabaza until the time of Uthman, where he died. So it was entirely possible for them to have met, and this is why the leading scholars relied on the continuity of this chain of transmission and graded it sahih.As for what Nasai (2410) narrated through the route of Abdullah (that is, Ibn Mubarak), from Asim, from Abu Uthman, from a man, who said, "Abu Dharr said, the Messenger of Allah ﷺ said," and then he related the hadith with the wording, "Whoever fasts three days of the month has completed the fasting of the month, or he has the reward of fasting the whole month." Although this indicates that there was a man between them, it may be that Abu Uthman first heard this from that man, who was certainly a companion, and then later had the opportunity to meet and hear directly from Abu Dharr himself, and so he narrated from him because they were both in Madinah.Perhaps Ibn Madini relied on this narration and thus denied that Abu Uthman had heard directly from Abu Dharr, without considering the other narrations. Moreover, the book of Ibn Madini may have been a draft that he never finished refining, which is why Abu Hasan bin Bara did not hear it directly from him. I have not come across any statement from any scholar denying that Abu Uthman Nahdi heard from Abu Dharr. Furthermore, Abu Uthman Nahdi was never described as practicing tadlis, so his use of "an" (from) should be taken as indicating direct transmission.As for the hadith of Abu Hurairah that Tirmidhi referred to, it has already been mentioned at the beginning of the chapter.