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2×2×2×3×3×3×5=1080, the least common multiple. We then first divide 8, 12 and 24 by 2, and place the quotients 4, 6, and 12 respectively under the numbers divided, at the same time placing below the line, together with the quotients, the numbers 9, 15, and 27, in which 2 is not factor. We then divide 4, 6, and 12 by 2; placing the quotients 2, 3, and 6, and the numbers 9, 15, and 27 below, as before. Finding that 2 will still divide two of the numbers, we divide 2 and 6 by 2, and omitting the quotient 1, we place the quotient 3 and the numbers 9, 3, 15, and 27 below, as before. Then, as 2 will no longer divide two of the numbers, and as they are all divisible by 3, we divide all by 3, and place the quotients 3, 5, and 9 below the line. Again, as 3 and 9 are divisible by 3, we divide and place the quotient 3 and the prime number 5 below the line. Lastly, as 3 and 5 are prime, we multiply these and all the divisors together, and have 1080 for the least common multiple.

By this method, we have amongst the divisors all the prime factors which are common to any two of the given numbers, and each is repeated as often as it is common factor in any two. Also, if any of these is still factor in one of the numbers below the line, that number is still retained, and is never again divided by the same factor. Therefore, each prime which is several times factor in any of the given numbers, will be just as many times factor in the multiple as in that number in which it is oftenest factor.

Again, as we have below the line all the numbers, whether prime or not, which have no common factor, it is evident that when we multiply these and the divisors together, the product is the product of all the prime factors of the given numbers, each taken in the highest power in which it is found in any one of them. Wherefore this product is a common multiple of all the given numbers.

Now as no prime number is found factor amongst the num

bers multiplied oftener than in some one of the given numbers: also, (180,) as no number can be a multiple of another, unless all the prime factors of that other enter into it as factors, it is evident that the product is the least common multiple. For, if we omit any one of the prime factors, (180,) the product must cease to be a multiple of the number in which that prime is oftenest factor.

184. Though the above rule is infallible, it will appear clumsy to one skilled in discovering the prime factors of numbers. For, by a mere inspection of the numbers 8, 9, 12, 15, 24, 27, we easily see that the only prime factors are 2, 3, and 5; that 8 is the highest power of 2; that 27 is the highest power of 3, and that 5 is only once factor: we, therefore, say, 8x27x5=1080, the least common multiple.

185. The least common multiple of several numbers is the greatest of those numbers; or, if not, it is a multiple of the greatest. Wherefore we may often readily find the multiple thus see if the greatest number is a multiple of all the others; if it is, it is the least common multiple. If it is not, try twice that number. If this does not succeed, try 3, 4, &c. times the greatest. Thus, if we would find the least common multiple of 6, 9, 18, 27, and 54, we see at once that it is 54. Again, if we would find the least common multiple of 2, 4, 6, 8, 12, 16, we first try if 16 is a common multiple of all the other numbers; it is not: we then try twice 16, which is 32; as this also fails, we try 3 times 16; and, as this succeeds, 48 is the required multiple.

Also, the least common multiple may often be conveniently found by selecting the greatest numbers that are prime to each other, and finding their product. Thus, if we would find the least common multiple of 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36, we select the numbers 8 and 9, which are prime to each other; and, having multiplied them together, we find that their product 72 contains all the given numbers, and is, consequently, (179,) their least common multiple.

If the greatest number, or any other, be a multiple of any of the given numbers, those numbers may, in operating by the general method, be omitted. Thus, if we would find the least common multiple of 3, 4, 6, 9, 12, 18, 27, 48, and 54, as 54 is a common multiple of 3, 6, 9, 18, and 27, we omit these numbers. Again, as 48 is a common multiple of 4 and 12, we omit these also. We therefore seek the least common

multiple of 48 and 54, and we find 432, which is also the least common multiple of all the given numbers.

Examples.

1. Required the least common multiple of 3, 5, 7, 4, and 11. Answer 4620.

To multiply numbers with facility, much often depends on the order in which they are taken. Thus, in the present example, without writing a figure, we first say, 4 times 5 is 20; and, leaving 0 till the last, we say 3 times 2 is 6, and 7 times 6 is 42. Then, (126,) for 11 times 42, as the sum of 4 and 2 does not exceed 9, we place their sum 6 between those figures, and have 462. Lastly, we place 0 on the right, and have 4620 for the product.

2. Required the least common multiple of 2, 5, 10, 11, and 55. Answer, 110. 3. Required the least common multiple of 5, 8, 9, 11, and 13. Answer, 51480. 4. What is the least common multiple of the nine digits?

Answer, 2520. 5. What is the least common multiple of 3, 17, 18, 51, and 54 ? Answer, 918. 6. What is the least common multiple of 6, 14, 39, 42, 78, and 91? Answer, 546. 7. What is the least common multiple of 18, 28, 33, 42, 48, 63, and 99 ? Answer, 11088. 8. What is the least common multiple of 6, 8, 12, 14, 16, 18, 21, and 22? Answer, 11088.

186. The number which divides the greater of two numbers and the less, will (172) divide their difference. Supposing the difference greater than the less number, the number which divides the less and the difference will also divide their difference. Again, supposing the difference still greater than the less, the number which divides the less number and this second difference will also divide their difference. Now, this continual subtraction of the less number is the same as the division of the greater number by the less. The final remainder must be less than the less number, and we might reason, as above, with respect to this remainder and the less number. Again, if there be still a remainder, the same reasoning may be pursued with respect to this remainder and the first remainder. Pursuing the same method, as a greater number cannot divide a less, the lust remainder which succeeds in dividing the preceding will be

the greatest common measure. Wherefore, to find the greatest common measure of two composite numbers, we have the following

Rule.

Divide the greater number by the less; if there is no remainder, the number divided by is the greatest common

measure.

If there is a remainder, divide the less number by this remainder, and if this division gives no remainder, the number last divided by is the greatest common measure.

If there is still a remainder, divide the first remainder by the second; and thus continue, always dividing the preceding remainder by the last, till the division becomes exact. The last divisor is the greatest common measure.

When the last divisor is a unit, the numbers are prime to each other.

Examples.

1. Required the greatest common measure of 9388 and 16429.

We proceed, according to the rule, thus:

9388) 16429 (1
9388

7041) 9388 (1
7041

2347) 7041 (3
7041

****

and have 2347, the last divisor, for the required measure. Because 2347 divides 7041, it will evidently divide 7041+ 2347, which is 9388. Again, because 2347 divides 7041 and 9388, it will divide their sum, which is 16429. Therefore 2347 is a common measure of 9388 and 16429.

No number greater than 2347 can be their common measure: for, if possible, let x be greater than 2347, and, at the same time, their common measure. Then, because x measures 9388 and 16429, it measures their difference 7041: also, because it measures 9388 and 7041, it measures their difference 2347; that is, a number greater than 2347 measures it, which is absurd. Therefore 2347 is the greatest common

measure.

2. Required the greatest common measure of 3675 and 5880. Answer, 735. 3. Find the greatest common measure of 4437 and 5899.

Answer, 17. 4. Find the greatest common measure of 23205 and 31395. Answer, 1365.

5. Required the greatest common measure of 46503 and 57546. Answer, 9.

6. Required the greatest common measure of 698 and 5343.

Answer,

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7. Required the greatest common measure of 34177 and 1012924.

OF RATIO.

Answer, 11.

187. In division we say that the greater the dividend is, the greater will the quotient be, and the less the dividend, the less the quotient, (165.) This relation of the dividend and quotient is called direct ratio.

Again, the greater the divisor is, the less will the quotient be; and the less the divisor, the greater the quotient. This relation of the divisor and quotient is called inverse ratio.

188. A simple ratio is the relation which one number bears to another, with respect to how often the one contains the other, or to what part the one is of the other; and is, in either case, expressed by that one divided by that other. Thus, if we would have the ratio of 3 to 5, or find what part 3 is of 5: as 1 unit is of 5, it is plain that 3 units must be ; that is to say, the ratio of 3 to 5 is 3 divided by 5. Again, if we would have the ratio of 5 to 3, or find how often 5 contains 3, this is evidently ğ.

A

Let A and B be any two homogeneous quantities—that is, quantities, the units of which are alike; then, if A be greater than B, shows how often A contains B: and, if A be less than B, shows what part A is of B; for, B being a whole or unit divided into equal parts, determines the value of these parts, and A is the number of the same parts which constitute the fraction Wherefore, the ratio of A to B is

and that of B to A is. The ratio of 8 to 4 is § =2, and the ratio of 4 to 8 is = 1.

Hence, every fraction expresses a ratio; namely, the ratio of the numerator to the denominator.

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