Numbers are divisible by

2, when they are even.

3, when the sum of their digits is divisible by 3.

4, when their two right-hand digits are divisible by 4. 5, when they have 5 or 0 in the units' place.

6, when they are even, and the sum of their digits is divisible by 3.

8, when their three right-hand digits are divisible by 8.

9, when the sum of their digits is divisible by 9.

10, when they have 0 in the units' place.

11, when the difference between the sum of the digits in the odd places and the sum of the digits in the even places is either 0, or is divisible by 11.

12, when the two right-hand digits are divisible by 4, and the sum of the digits divisible by 3.

For the numbers 7 and 13 we may observe that since 1001=7 × 11 × 13, it follows that if the number of thousands differs from the number of units by 0, or a multiple of 7 or 13, the number is divisible by 7 or 13.

[It should be explained that the numbers are to be read thus units, tens, hundreds, of units; units, tens, hundreds of thousands; units, tens, hundreds of millions; &c. Thus in the number 31178, there are 178 units and 31 thousands; and 178-31=147=7×21; whence the number is divisible by 7. In the number 153517 there are 517 units and 153 thousands ; and 517-153=364=7 × 52=7 × 13 × 4, whence the number is divisible by both 7 and 13.]

All prime numbers, except 2 and 5, have either 1, 3, 7, or 9 in the place of units; but it is not conversely true that all numbers having 1, 3, 7, or 9 in the place of units are prime.

We can now proceed to find the prime factors of any number. [Obs. We must first explain that when a number is multiplied into itself any number of times, the product is called a power of the number; above the number and to the righthand of it is written a small figure, which denotes the number of factors that produces the power; and this figure is called the Index: thus 2 × 2 is called 2 squared, or 2 raised to the second power, and is written 22; 2 × 2 × 2 is called 2 cubed, or 2 raised to the third power, and is written 23; and so on.]

The method of decomposing or resolving any number into its prime factors is as follows: Divide the given number successively, and as often as possible, by each of the prime numbers, 2, 3, 5, 7, &c., beginning with the lowest prime divisor that will measure the given number: when the last quotient is prime, this prime quotient and the several divisors which have been used are the prime factors required.

Ex. 1. Decompose 4550 into its prime factors.

214550

52275

5455

7 91

13

And as the last quotient 13 is prime, the required prime factors are 2 × 5 × 5 × 7 × 13, which may be written 2 × 52 × 7 × 13. Ex. 2. Decompose 11088 into its prime factors.

Hence the prime factors required are 2a × 32 × 7 × 11.

As in this example we can see by inspection that 11088 is divisible by 8, since the two right-hand digits are divisible by 8, we might have divided at once by 8, or by 23; and then, after the next division by 2, we might have seen that the quotient 693 was divisible by 9, or by 32; and the operation in a shortened form might have stood thus:

E

By decomposing numbers into their prime factors we may find either their G. C. M. or their L. C. M.

Take for instance, the numbers 1260, 10584, 12960; decomposed into their prime factors, they become respectively 22 × 32 × 5 × 7, 23 × 33 × 73, 25 × 3a × 5; and of these the only factors which are common to all, are 22 and 32; whence the greatest number which will measure them all, or the G.C.M., is 22 × 32, or is 4 × 9, or 36.

On the other hand, the least number which will contain all these prime factors must evidently contain the highest powers of each of them; that is, the L. C. M. must contain 25, 34, 5 and 72; and therefore the L. C. M. is the product 25 × 34 × 5 × 73, or is 635040.

From this we deduce the following rule for finding the G. C. M. or the L. C. M. of several numbers. Decompose the given numbers into their prime factors: multiply together the lowest powers of those factors which are common to all; the product so formed will be the G. C. M. of the given numbers. Multiply together the highest powers of all the factors that occur; the product so formed will be the L. C. M. of the given numbers.

3. 7648, 13384, 63096.

4. 12562, 4568, 5139, 8565.

5. 4230, 141000, 95175, 3760, 27636. 6. 22578, 13144, 1113.

VI. Decompose into their prime factors, and thence find the G. C. M. and the L. C. M. of the numbers,

1. 17725554, 1054872, and 2406096.

2. 340362, 37818, and 7147602.

VII. Find the L. C. M. of 4, 12, 16, 20, and 36; also of 5, 7, 16, 28, 48, and 21.

VIII. Find the greatest number which will divide 398, and 442, leaving as remainders respectively 7 and 5.

IX. Required the least number which can be divided by 7, 12, 15, and 24, with a remainder 3 in every case.

X. Required the least number which when divided by 5, 8 and 9, gives in every case the remainder 2.

XI. Find the greatest number which will divide 6332, and 23999, leaving as remainders 5 and 2 respectively.

XII. Find the greatest number by which when 3863 and 4769 are divided, the respective remainders are 3 and 1.

XIII. Which of the numbers 16137, 41481, 47032, 6809517, 998216, are divisible by 8, 9, 11, or 12?

91189, 71799, 51982, and 7389501522 for the factors 7 and 13.

1

52

CHAPTER V.

ENGLISH AND FOREIGN WEIGHTS AND MEASURES.

§ 41. As regards the different denominations of money, &c. we have hitherto only assumed that it is known that 4 farthings make a penny, that 12 pence make a shilling, and that 20 shillings make a pound. But it is a great incon·venience in our system of money, weights, and measures, that there is nothing uniform about it. The most convenient system is, no doubt, the decimal: and from time to time there have been attempts made to introduce such a system into England, and to adapt our coins, as well as our weights and measures, to those in general use on the Continent. But though the subject has been a good deal discussed, little progress has hitherto been made; the practical difficulties in carrying out alterations in long-established customs and habits. have appeared so formidable, that they have been sufficient to deter the legislature from taking decisive action in the matter. We must therefore be content at present to use the following "Tables," in which the measures in common use are given. But inasmuch as the whole subject is attracting more and more attention, especially after the International Conference at Paris in 1867, (to which further allusion will be made below,) it appears only proper to subjoin to our own tables an account of the metric system of France; and then to give some account of the various schemes which have been proposed both for making our coinage decimal, and likewise for carrying into practical effect the project of a universal coinage for all nations.

The English tables in ordinary use are as follows: