4. What is the greatest common measure of 376 and 940:? Ans. 188.

5. What is the greatest common measure of 144 and 240? Ans. 48.

6. What is the greatest common measure of 1376 and 9402 ? Ans. 2.

167. To find the greatest common measure of three or more

numbers.

RULE I. Find the greatest common measure of any two of the given numbers by the last rule.

II. Find the greatest common measure of this common measure and another of the given numbers by the last rule.

III. Find the greatest common measure of this last common measure and another of the given numbers, and so on until all the given numbers have been taken; the last common measure is the greatest common measure of all the given numbers *.

7. Required the greatest common measure of 32, 48, and 68.

OPERATION.

First 32)48(1

32

16)32(2

32

Then 16)68(4 64

4)16(4)
16

Ans. 4.

Explanation.

First I find the greatest common measure of 32 and 48, (by Art. 166,) which is 16.

Then I find the greatest common measure of 16 and the remaining number 68, which is 4; therefore 4 is the greatest common measure of the three given numbers 32, 48, and 68, as was required.

k Having found the greatest common measure of two of the given numbers, if this measures the third, it will evidently be the greatest common measure of all the three; but if not, then it is equally evident that the greatest common measure of the said common measure and of the third number will be the greatest common measure of the three given numbers; and in the same manner the greatest common measure of four or more numbers may be accounted for.

8. What is the greatest common measure of 144, 216, and 324?

The greatest common measure of 144 and 216, by Art, 166, is 72, and the greatest common measure of 72 and 324 is 36, the greatest common measure required.

9. What is the greatest common measure of 88, 132, 154, and

165 ?

The greatest com, meas. of 88 and 132 is 44.

of 44 and 154 is 22.

of 22 and 165 is 11, the answer req. 10. What is the greatest common measure of 72, 120, and 132? Ans. 12.

11. What is the greatest common measure of 376, 940, 1034, and 1081 ? Ans. 47.

12. Required the greatest common measure of 100, 200, 350, 425, and 505. Ans. 5.

168. To find the least common multiple of two given numbers. RULE I. Find the greatest common measure of the two given numbers, by Art. 166.

II. Multiply the two given numbers together, and divide the product by the greatest common measure; the quotient is the least common multiple required1.

13. What is the least common multiple of 12 and 18 ?

For the greatest common measure of two numbers will evidently measure their product; and if a number greater than the greatest common measure be assumed, then one of the two numbers being divided by it, the quotient will be a fraction, and the other given numb. being multiplied by this fraction, the product will not be a multiple of that number; whence the rule is plain.

14. What is the least common multiple of 48 and 72 ? Their greatest common measure (by Art. 166.) is 24, their product 48 × 72 = 3456.

and

15. To find the least common multiple of 30 and 40.

Greatest common measure 10.

16. What is the least common multiple of 15 and 20? Ans. 60. 17. Find the least common multiple of 60 and 84. Ans. 420. 18. Required the least common multiple of 108 and 162. Ans. 324.

169. To find the least common multiple of three or more

numbers".

RULE I. Find the least common multiple of any two of the numbers by the last rule.

II. Find the least common multiple of this multiple and another of the numbers, and it will be the answer for three numbers.

III. Find the least common multiple of this last multiple and another of the numbers, and it will be the answer for four numbers.

IV. Proceed in this manner until you have obtained the least common multiple of all the given numbers.

m The rule given by Mr. Bonnycastle is as follows:

"1. Divide by any number that will divide two or more of the given numbers without remainder, and set the quotients, together with the undivided numbers, in a line beneath."

"2. Divide the second line as before, and so on till there are no two numbers that can be divided; then the continued product of the divisors, quotients," and undivided numbers, "will give the multiple required."

Thus, to find the least common multiple of 4, 10, and 15.

19. What is the least common multiple of 4, 10, and 15? The least common multiple of 4 and 10 (by Art. 168) is

Therefore 60 is the least common multiple required. 20. What is the least common multiple of 12, 16, and 30?

Least common multiple of 12 and 16 is

Least common multiple of 48 and 30 is

12 x 16
4

= 48.

21. Find the least common multiple of the nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9.

Here, because 8 is a multiple of 1, 2, and 4, every multiple of 8 will be some multiple of 1, 2, and 4; in like manner 6 being a multiple of 3, every multiple of 6 will be some multiple of 3; wherefore in the operation the numbers 1, 2, 3, and 4, may be omitted, as being aliquot parts of some of the other numbers. To find the least common multiple therefore of all the nine digits, we have only to find that of 5, 6, 7, 8, and 9.

The least common multiple of 5 and 6 is

5 × 6

= 30; of

1

22. What is the least common multiple of 3, 4, and 8? Ans. 24.

23. What is the least common multiple of 3, 5, 8, and 10? Ans. 120.

24. What is the least common multiple of 3, 8, 14, and 22 ? Ans. 1848.

170. To reduce a fraction to its lowest terms.

RULE I. Find the greatest common measure of the numerator and denominator by Art. 166.

II. Divide both terms of the fraction by the greatest common measure, and the quotients will be the numerator and denomi

nator respectively of the fraction which expresses the lowest terms of the given fraction".

Secondly, divide both terms of the given fraction by the greatest

The greatest common measure by Art. 166 is 12.

n Two fractions are equal to each other when the numerator of one has to its denominator the same ratio which the numerator of the other has to its denominator, as is plain from the definition of a fraction; hence it follows that the same fraction may be expressed in a great variety of ways; thus, is the same as oror or for 4 or 4, &c. &c. ; this being premised, the above rule teaches to find the least numbers possible that will express any given fraction; if a fraction be not in its lowest terms, both terms must evidently be divided, and they must both be divided by the same number, otherwise the terms would not be proportionals, and therefore the resulting fraction would not equal the given one; (see note on Art. 165.): moreover this divisor must be the greatest possible, (viz, the greatest common measure of both terms,) otherwise the quotients will not be the least possible: whence the rule is plain.