Elementary arithmetic, with brief notices of its history... by Robert Potts.

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remainder; except when the sum of the latter exceeds the next higher
unit; in that case both the quotient and remainder must be increased
by unity.'
Any number can be divided by 5, 25, 125, &c., by multiplying the,dividend by 2, 4, 8, &c., and marking off towards the left from the
unit's place, one, two, -three, &c., figures. The figures on the left
will be the quotient, and those on the right will be 2, 3, 4, &c., times
the respective remainders.2
11. PROB. To divide a numnber by tfhe factors of the divisor, and to
determine the correct remainder after the division.
Divide 109958 by 5, 7, 11, the factors of 385, and determine the
Qcorrect remainder.
5)109958
7)21991 - 3 first remainder.
11)3141 - 4 second remainder.
285 - 6 third remainder.
The quotient is 285, with remainders 3, 4, 6, after the divisions by
5, 7, 11 respectively.
Every unit in the 2nd line is 5 times each unit in the 1st line.
3rd 7 ', 2nd line.
4th,,11,, 3rd line.
lHence3, the remainder from the 1st line, is 3 units of the 1st line.
4,,, 2nd,, 5 times 4 or 20,,
6,,, 3rd,, 7times5times6or210,
The sum of these partial remainders is 233, the whole remainder.
The process may be verified by dividing 109958 by 385.
The product of two numbers is the same, whichever of the two
be made the multiplier, as also of any number of factors, in whatever
order the numbers may be multiplied. It follows, that if a dividend be
divided by several divisors in succession, the quotient and the remainder,are the same, in whatever order the separate divisions are performed.
1 Ex. Divide 65874 by 99.
100)658,74
6,58
6
665,39
Here the sum of the partial remainders is 138, and both the quotient and re.
-mainder must be increased by unity. The truth of the process may be verified by
-performing the division of 65874 by 99 in the ordinary way.
2 George Suffield, M.A., late Fellow of Clare College, published a small tract
entitled, " Synthetic Division in Arithmetic, with some introductory remarks on
the periods of circulating decimals.'" This method is useful for obtaining quotients
when a very large number of figures is required. It contains some brief and ingenious methods for converting ordinary fractions into repeating decimals..faxcminillan and Co., Cambridge and London, 1863.