Remainder arithmetic formally appeared in Egypt about 4,000 years ago. The Akhmim Wooden Tablet (AWT), circa 1950 BCE, theoretically defined a remainder arithmetic application within a weights and measures context. Prior to 2,000 BCE, remainder arithmetic may have been intuitively used in Egyptian calendars. The AWT scribe theoretically partitioned a volume unit, named hekat, into cubit-cubit units by beginning with a hekat unity written as (64/64). Two classes of Hekat sub-units were created. The first divided (64/64) by rational numbers n in the AWT. The second divided m, a small set of hekat multiples, by n in the Rhind Mathematical Papyrus (RMP), circa 1650 BCE.

The AWT allowed divisors n to equal 3, 7, 10, 11 and 13. The RMP allowed divisors n in the range 1/64 < n < 64. Answers to hekat unity divisons were written in binary quotients, and (5R/n)*1/320 Egyptian fraction remainders (R).

The second remainder arithmetic method was documented in 1650 BCE, though the method may be older. The RMP also divided multiples of a hekat m to be set to 20, 10, 5, 4, 1/10, 1/20, 1/64, 1/320 and other values. Divisors were allowed to be any rational number. This method created sub-units in the form m/n "name". The volume unit used one-part integer quotients, and non-scaled Egyptian fraction remainders named hin, dja, ro, and so forth. For example, the 1/10 hin unit was written as 10/n hin, the dja unit was written as 64/n dja, and the ro unit was written as 320/n ro. RMP 81 lists a table of 29 divisors limiting n to 1/64 < n < 64 with first method's data in one column, and equivalent second method data in another column.

Scholars first reported the hekat unity method in 1906 (Daressy). A proof side of the first method was reported in 2002 (Vymazalova). The m/n "name" method correctly scaled the dja in 2005 (Pemmerening).

The AWT's five scaled examples are sufficient to define the oldest known remainder arithmetic method. The AWT scribe divided a hekat unity, written as: divided by integers 3, 7, 10, 11, and 13. Answers were converted to two-part binary quotients, and Egyptian fraction remainders. The two-part answers were proven. Proof was achieved by multiplying the quotient and remainder answer by the initial divisor (3, 7, 10, 11 or 13). The AWT scribe returned the answers to the initial hekat unity. Hana Vymazalova published the proof side of this set of facts in 2002, with others reporting the AWT's remainder arithmetic properties in 2005.

The AWT text was first published in 1901, and first analyzed by Daressy in 1906. Daressy analyzed the text line by line, correctly reported three hekat divisions: divisors 3, 7, and 10. Daressy improperly reported n = 11 and n = 13 divisons and the proof side of all five problems. The hekat unity division method remained unresolved until teh proof side of the topic was corrected by Hana Vymazalova in 2002.

Hana Vymazalova, a Charles U., Prague grad student published AWT hekat problems by correcting Daressy's two 1906 errors. Vymazalova proved that all five two-part answers had been returned by the AWT scribe to a hekat unity . Vymazalova's paper did not suggest a theoretical context. Yet, she clearly showed that the AWT scribe and other MK scribes recorded binary quotients, and scaled Egyptian fraction remainders in an innovative manner that had not been present in the Old Kingdom.

Concerning the second remainder arithmetic method, over 2,000 Middle and New Kingdom (NK) medical prescriptions were written in a method. Several NK units had been garbled by translators, much as Daressy had garbled two AWT answers. In 2002, a German graduate student, Tanja Pommerening, published a dja hekat unit properly scaled to 1/64 of a hekat, correcting a unit that had confused modern physicians trying to reconstitute ancient Egyptian medicines.

Dr. Tanja Pommerening's 2005 PhD discussed the dja as a healing unit correcting any rounded-off weights and measures Horus-Eye series. The "healed" definition is recorded in hieratic and hieroglyphic symbolisms. Pommerening depicted the dja unit as a hekat sub-unit equal to 1/64 of a hekat. The dja was written in a one-part remainder arithmetic system. The "healed" 1/64 corrected Old Kingdom rounded off 1/64 measurements may be written in other weights and measures system(s), i.e. balance beam weights.

Details of the 4,000 year old methods can be found on Akhmim Wooden Taqblet.