Greatest Common Divisor 3.4.3 47.
Division by a Mixed Number.-To divide by a mixed number, when the quotient is seen to be large, it usually saves time to express the divisor as either a simple fraction or a decimal of a unit of one of the denominations.
Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J.
Methods of Division.-What are described as different methods of division (by a single divisor) are mainly different methods of writing the successive figures occurring in the process.
They there fore study that the numbers of teeth in each pair of wheels whici work together shall either be prime to each other, or shall hav their greatest common divisor as small as is consistent with velocity ratio suited for the purposes of the machine.
The term division is therefore used in text-books to describe the two processes described in §§ 38 and 39; the product mentioned in § 34 is the dividend, the number or the unit, whichever is given, is called the divisor, and the unit or number which is to be found is called the quotient.
In long division the divisor is put on the left of the dividend, and the quotient on the right; and each partial product, with the remainder after its subtraction, is shown in full.
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.
Algebraical division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (ï¿½ 28 (iv.)), and are arranged in descending or ascending powers of x.
So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and ii shall be either 1, or as small as possible.
If we resolve two numbers into their prime factors, we can find their Greatest Common Divisor or Highest Common Factor (written G.C.D.
(iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple.
If the greatest common divisor of N and n be d, a number less than n, so that n=md, N~Md; then a=mN=Mn=Mmd; b=M; c=m.
In determining the rate of a " dividend "), approximate expression of the divisor in terms of the largest unit is sufficient.
If n is a divisor of N,