100. If two numbers be divisible by a third, and if the quotients be again divisible by a fourth, that third is not the greatest common measure. For example, 360 and 504 are both divisible by 4. The quotients are 90 and 126. Now 90 and 126 are both divisible by 9, the quotients of which division are 10 and 14. By (87), dividing a number by 4, and then dividing the quotient by 9, is the same thing as dividing the number itself by 4 × 9, or by 36. Then, since 36 is a common measure of 360 and 504, and is greater than 4, 4 is not the greatest common measure. Again, since 10 and 14 are both divisible by 2, 36 is not the greatest common measure. It therefore follows, that when two numbers are divided by their greatest common measure, the quotients have no common measure except 1 (99). Otherwise, the number which was called the greatest common measure in the last sentence is not so in reality.
101. To find the greatest common measure of three numbers, find the g. c. m. of the first and second, and of this and the third. For since all common divisors of the first and second are contained in their g. c. m., and no others, whatever is common to the first, second, and third, is common also to the third and the g. c. m. of the first and second, and no others. Similarly, to find the g. c. m. of four numbers, find the g. c. m. of the first, second, and third, and of that and the fourth.
102. When a first number contains a second, or is divisible by it without remainder, the first is called a multiple of the second. The words multiple and measure are thus connected: Since 4 is a measure of 24, 24 is a multiple of 4. The number 96 is a multiple of 8, 12, 24, 48, and several others. It is therefore called a common multiple of 8, 12, 24. 48, &c. The product of any two numbers is evidently a common multiple of both. Thus, 36 × 8, or 288, is a common multiple of 36 and 8. But there are common multiples of 36 and 8 less than 288; and because it is convenient, when a common multiple of two quantities is wanted, to use the least of them, I now shew how to find the least common multiple of two numbers.
103. Take, for example, 36 and 8. Find their greatest common measure, which is 4, and observe that 36 is 9 × 4, and 8 is 2 × 4. The quotients of 36 and 8, when divided by their greatest common measure, are therefore 9 and 2. Multiply these quotients together, and multiply the product by the greatest common measure, 4, which gives 9 × 2 × 4, or 72. This is a multiple of 8, or of 4 × 2 by (55); and also of 36 or of 4 × 9. It is also the least common multiple; but this cannot be proved to you, because the demonstration cannot be thoroughly understood without more practice in the use of letters to stand for numbers. But you may satisfy yourself that it is the least in this case, and that the same process will give the least common multiple in any other case which you may take. It is not even necessary that you should know it is the least. Whenever a common multiple is to be used, any one will do as well as the least. It is only to avoid large numbers that the least is used in preference to any other.
When the greatest common measure is 1, the least common multiple of the two numbers is their product.
The rule then is: To find the least common multiple of two numbers, find their greatest common measure, and multiply one of the numbers by the quotient which the other gives when divided by the greatest common measure. To find the least common multiple of three numbers, find the least common multiple of the first two, and find the least common multiple of that multiple and the third, and so on.
EXERCISES.
| Numbers proposed. | Least common multiple. |
|---|---|
| 14, 21 | 42 |
| 16, 5, 24 | 240 |
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 2520 |
| 6, 8, 11, 16, 20 | 2640 |
| 876, 864 | 63072 |
| 868, 854 | 52948 |
A convenient mode of finding the least common multiple of several numbers is as follows, when the common measures are easily visible: Pick out a number of common measures of two or more, which have themselves no divisors greater than unity. Write them as divisors, and divide every number which will divide by one or more of them. Bring down the quotients, and also the numbers which will not divide by any of them. Repeat the process with the results, and so on until the numbers brought down have no two of them any common measure except unity. Then, for the least common multiple, multiply all the divisors by all the numbers last brought down. For instance, let it be required to find the least common multiple of all the numbers from 11 to 21.
| 2, 2, 3, 5, 7) | 11 12 13 14 15 16 17 18 19 20 21 |
| 11 1 13 1 1 4 17 3 19 1 1 |