traction operation, is the sum of the remainder and the subtrahend, it is plain that every common factor of the remainder and the subtrahend is a factor of the minuend.

The Application of the Method.-We pass now to the application, and shall take the numbers 851 and 1073. The difficulty has been that these numbers are large, and in reply to the question, What smaller number will have in it any common factor that 851 and 1073 may have? there might be expected the answer, 1073851. But there must be an examination of this statement.

If 851 and 1073 have a common factor, this factor will also measure 222; and if 222 and 851 have a common factor, this factor will measure 1073. Thus the greatest common measure of 851 and 1073 is a factor of 222, and the greatest common measure of 851 and 222 is a factor of 1073. Therefore the greatest common measure of 851 and 222 is the greatest common measure of 851 and 1073. It will now be easy to show that if 222, or 2 times 222, or 3 times 222, be taken from 851, 222 and this remainder will have for greatest common factor the greatest common factor of 851 and 222, and the advantage in taking from 851, 3 times 222 is apparent.

It will be easy to follow this out through the suc

cessive steps:

37 divides 185 exactly, and is thus the greatest common measure of 185 and 37; so that 37 is the greatest common measure of

and a conviction will be added to the proof. Then the identity of the work with the following may be shown:

851)1073(1
851

222)851(3

666

185)222(1

185

37)185(5
185

We see now that a definite method has been evolved, and when the class has been exercised in applying it, it may be well to explain certain artifices by means of which the work may be shortened, or exhibited in a neater form. For example, the work of finding the

greatest common measure of 851 and 1073, as given above, may be presented as follows:

Or the work might be conveniently arranged as in the following example: Find the greatest common measure of 158938 and 531206.

The quotients appear in the middle column, and the work explains itself.

It is to be observed that if any common factor is easily discoverable in the two given quantities, it is better first to divide both quantities by the common factor. If, also, a prime factor is found in only one of the quantities which are in operation for the greatest common measure, it may be struck out. In the last example, for instance, the first remainder is divisible by 8, while the corresponding number on the other side (the first divisor) is divisible by 2. We may therefore divide this number by 2 and the other by 8, reserving 2 as

part of the required common measure. These factors being removed, we operate with the quotients, 79469 and 6799. The latter divides the former with remainder 4680; this, it is obvious, has the factors 40, 13, 9. Hence, if the two original quantities have a common factor, it is 13 X 2-a result obtained by the actual work.

This study of the measures of numbers suggests classifications of numbers. Numbers may be (1) even or odd, according as they do or do not contain 2 as a factor; (2) composite or prime, according as they are or are not resolvable into simpler factors.

Two numbers may have no factors in common, though each of them may be composite; they are then said to be prime to each other. It will be supposed that the class is familiar with these classifications and definitions before proceeding to a study of least common multiple.

LEAST COMMON MULTIPLE.

In the presentation of least common multiple, it is necessary as indeed it always is in the introduction of a new process-first to bring out clearly the essential facts and ideas upon which the process rests. Here the pupil must first get a clear idea of the terms multiple, common multiple, least common multiple. A factor (or measure) of 15 is 3; 15 is called a multiple of 3; it represents the quantity that 3 exactly measures. The pupil will now be asked to name different multiples of 3, say, and will see that he may name or write down as many as he chooses. Then, if a series of multi

ples of 2 be written down, so that we have the two series:

3, 6, 9, 12, 15, 18, 21, 24, . . .

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,

he will see that there are numbers which are at the same time multiples of 2 and 3, and are therefore common multiples of 2 and 3. These common multiples are, here, 6, 12, 18, . . . Then, because we have started with the smallest multiples of the numbers, 6 is the smallest or least common multiple of 2 and 3. At this point the pupil can hardly fail to see that the second common multiple is 66, the third is 6+6+6, etc.; in other words, that all the common multiples of two numbers are formed by repeating as an addend the least common multiple. He can then be led to see the reason for this, viz., that (referring to the foregoing example), in order to get a common multiple of 2 and 3 larger than 6, it will be necessary to add to 6 a common multiple of 2 and 3, so that 6 is the least number that can be used.

Numbers Prime to Each Other.-A necessary step preliminary to teaching the formal process is the bringing out of the fact that the least common multiple of two numbers prime to each other is their product. For example, take the numbers 5 and 7: a common multiple must have 5 as a factor and 7 as a factor; it is, therefore, 5 multiplied by another factor. But since the multiple contains 7, and since 7 is prime to 5, the other factor of the multiple must contain 7. Hence, since the smallest multiple of 7 is 7, the least common multiple of 5 and 7 is 5 × 7. Similarly, a common multiple of 4 and 9 is a multiple of 4, and is therefore 4 multiplied