The important thing to notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that " if equals are taken from equals the remainders are equal "; the fact being that the two statements are merely different ways of expressing the same relation.
Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t,.
The principal ruins of the town of Pasargadae at Murghab are a great terrace like that of Persepolis, and the remainders of three buildings, on which the building inscription of Cyrus, "I Cyrus the king the Achaemenid" (sc. " have built this"),, occurs five times in Persian, Susian and Babylonian.
Some of them are relics of the longitudinal moraines of the ice-sheet, and they run north-west to south-east, parallel to the striation of the rocks and to the countless parallel troughs excavated by the ice in the hard rocks in the same direction; while the Lojo As, which runs from HangOudd to Vesi-jarvi, and is continued farther east under the name of Salpausellia, parallel to the shore of the Gulf of Finland, are remainders of the frontal moraines, formed at a period when the ice-sheet remained for some time stationary during its retreat.
At the foot of the rock are the remainders of some other sculptures (quite destroyed), the fragments of a Greek inscription of the Parthian prince Gotarzes (A.D.
Also, if the remainders when two numbers are divided by 9 are respectively a and b, the remainder when their product is divided by 9 is the same as the remainder when a.b is divided by 9.
In short division the divisor and the quotient are placed respectively on the left of and below the dividend, and the partial products and remainders are not shown at all.
The second, which is not essential to the method, is that the remainders are shown, but not the partial products; the remainders being obtained by working from the right, and using complementary addition.
The act of presentation (Vorstellen) then becomes partly transformed into an effort, and its product, the idea, becomes in the same proportion less and less intense till a position of equilibrium is reached; and then at length the remainders coalesce.
In incomplete partition the quotient is 3, and the remainders 11 and 17 are in effect disregarded; if, after finding the quotient 3, we want to know what remainder would be produced by'a direct division, the simplest method is to multiply 3 by 2 4 0 and subtract the result from 935.