3. What is the greatest common divisor of 336 and 812? Ans. 28.

4. What is the greatest common divisor of 407 and 1067? 5. What is the greatest common divisor of 825 and 1372? 6. What is the greatest common divisor of 2041 and 8476? Ans. 13. 7. What is the greatest common divisor of 3281 and 10778? 8. Find the greatest common divisor of 22579, and 116939. 9. What is the greatest common divisor of 49373 and 147731? Ans. 97. 10. What is the greatest common divisor of 1005973 and 4616175?

11. Find the greatest common divisor of 292, 1022, and 1095. Ans. 73. 12. What is the greatest common divisor of 4718, 6951, and 8876? Ans. 7. 13. Find the greatest common divisor of 141, 799, and 940. 14. What is the greatest common divisor of 484391 and 684877? Ans. 701.

15. A farmer wishes to put 364 bushels of corn and 455 bushels of oats into the least number of bins possible, that shall contain the same number of bushels without mixing the two kinds of grain; what number of bushels must each bin hold?

Ans. 91.

16. A gentleman having a triangular piece of land, the sides of which are 165 feet, 231 feet, and 385 feet, wishes to inclose it with a fence having pannels of the greatest possible uniform length; what will be the length of each pannel?

17. B has $620, C $1116, and D $1488, with which they agree to purchase horses, at the highest price per head that will allow each man to invest all his money; how many horses can each man purchase? Ans. B 5, C 9, and D 12. 18. How many rails will inclose a field 14599 feet long by 10361 feet wide, provided the fence is straight, and 7 rails high, and the rails of equal length, and the longest that can be used?

Ans, 26880.

LEAST COMMON MULTIPLE.

151. A Multiple is a number exactly divisible by a given. number; thus, 20 is a multiple of 4.

NOTES.1. A multiple is necessarily composite; a divisor may be either prime. or composite.

2. A number is a divisor of all its multiples and a multiple of all its divisors. 152. A Common Multiple is a number exactly divisible by two or more given numbers; thus, 20 is a common multiple of 2, 4, 5, and 10.

153. The Least Common Multiple of two or more numbers is the least number exactly divisible by those numbers; thus, 24 is the least common multiple of 3, 4, 6, and 8.

154. From the definition it is evident that the product of two or more numbers, or any number of times their product, must be a common multiple of the numbers. Hence, A common multiple of two or more numbers may be found by multiplying the given numbers together.

155. To find the least common multiple.

FIRST METHOD.

From the relations of multiple and divisor we have the following properties:

I. A multiple of a number must contain all the prime factors of that number.

II. A common multiple of two or more numbers must contain all the prime factors of each of those numbers.

III. The least common multiple of two or more numbers must contain all the prime factors of each of those numbers, and no other factors.

1. Find the least common multiple of 63, 66, and 78.

We here have all the prime factors, and also all the factors of 66 except 11. Annexing 11 to the series of factors,

2 x 3 x 13 x 11,

and we have all the prime factors of 78 and 66, and also all the factors of 63 except one 3, and 7. Annexing 3 and 7 to the series of factors,

2 x 3 x 13 x 11 × 3 × 7,

and we have all the prime factors of each of the given numbers, and no others; hence the product of this series of factors is the least common multiple of the given numbers, (III).

From this example and analysis we deduce the following

RULE. I. Resolve the given numbers into their prime factors. II. Multiply together all the prime factors of the largest number, and such prime factors of the other numbers as are not found in the largest number, and their product will be the least common multiple.

NOTE. When a prime factor is repeated in any of the given numbers, it must be taken as many times in the multiple, as the greatest number of times it appears in any of the given numbers.

EXAMPLES FOR PRACTICE.

1. Find the least common multiple of 60, 84, and 132.

Ans. 4620. 2. Find the least common multiple of 21, 30, 44, and 126. Ans. 13,860.

3. Find the least common multiple of 8, 12, 20, and 30. 4. Find the least common multiple of 16, 60, 140, and 210. Ans. 1,680.

5. Find the least common multiple of 7, 15, 21, 25, and 35. 6. Find the least common multiple of 14, 19, 38, 42, and 57. Ans. 798.

7. Find the least common multiple of 144, 240, 480, 960.

SECOND METHOD.

156. 1. What is the least common multiple of 4, 9, 12, 18, and 36?

2 × 2 × 3 × 3 = 36 Ans.

ANALYSIS. We first write the given numbers in a series with a vertical line at the left. Since 2 is a factor of some of the given numbers, it must be a factor of the least common mul

tiple sought, (155, III). Dividing as many of the numbers as are divisible by 2, we write the quotients, and the undivided number, 9, in a line underneath. Now, since some of the numbers in the second line contain the factor 2, the least common multiple must contain another 2, and we again divide by 2, omitting to write any quotient when it is 1. We next divide by 3 for a like reason, and still again by 3. By this process we have transferred all the factors of each of the numbers to the left of the vertical; and their product, 36, must be the least common multiple sought, (155, III).

2. What is the least common multiple of 20, 12, 15, and 75?

that is divisible by either of these factors or by their product; thus, we divide 20 by both 2 and 5; 12 by 2; 15 by 5; and 75 by 5. We next divide the second line in like manner by 2 and 3; and afterward the third line by 5. By this process we collect the factors of the given numbers into groups; and the product of the factors at the left of the vertical is the least common multiple sought.

3. What is the least common multiple of 7, 10, 15, 42, and 70?

3,7

OPERATION.

15.. 42.. 70

2,5 5 2..10

ANALYSIS. In this operation we omit the 7 and 10, because they are exactly contained in some of the other given numbers; thus, 7 is contained in 42, and 10 in 70; and whatever will contain 42 and 70 must contain 7 and 10. Hence we have only to find the least common multiple of the remaining numbers, 15, 42, and 70.

3 x 7 x 2 x 5

=

210, Ans.

From these examples we derive the following

RULE. I. Write the numbers in a line, omitting such of the smaller numbers as are factors of the larger, and draw a vertical line at the left.

II. Divide by any prime factor or factors that may be contained in one or more of the given numbers, and write the quotients and undivided numbers in a line underneath, omitting the 1's.

III. In like manner divide the quotients and undivided numbers, and continue the process till all the factors of the given numbers have been transferred to the left of the vertical. Then multiply these factors together, and their product will be the east common multiple required.

NOTE.

We may use a composite number for a divisor, when it is contained in all the given numbers.

EXAMPLES FOR PRACTICE.

1. What is the least common multiple of 15, 18, 21, 24, 35, 36, 42, 50, and 60?

Ans. 12600. 2. What is the least common multiple of 6, 8, 10, 15, 18, 20, and 24 ?

Ans. 360. 3. What is the least common multiple of 9, 15, 25, 35, 45, and 100? Ans. 6300. 4. What is the least common multiple of 18, 27, 36, and 40 ? 5. What is the least common multiple of 12, 26, and 52? 6. What is the least common multiple of 32, 34, and 36? Ans. 4896. 7. What is the least common multiple of 8, 12, 18, 24, 27, and 36?

8. What is the least common multiple of 22, 33, 44, 55, and 66?

9. What is the least common multiple of 64, 84, 96, and 216? 10. If A can build 14 rods of fence in a day, B 25 rods, C 8 rods, and D 20 rods, what is the least number of rods that will furnish a number of whole days' work to either one of the four men? Ans. 1400.