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10. A farmer had 231 bushels of wheat, and 273 bushels of oats, which he wished to put into the least number of bins containing the same number of bushels, without mixing the two kinds; what number of bushels must each bin hold?

Ans. 21.

11. A village street is 332 rods long; A owns 124 rods front, B 116 rods, and C 92 rods; they agree to divide their land into equal lots of the largest size that will allow each one to form an exact number of lots; what will be the width of the lots? Ans. 4 rods.

12. The Erie Railroad has 3 switches, or side tracks, of the following lengths: 3013, 2231, and 2047 feet; what is the length of the longest rail that will exactly lay the track on each switch? Ans. 23 feet.

13. A forwarding merchant has 2722 bushels of wheat, 1822 bushels of corn, and 1226 bushels of beans, which he wishes to forward, in the fewest bags of equal size that will exactly hold either kind of grain; how many bags will it take? Ans. 2885.

14. A has 120 dollars, B 240 dollars, and C 384 dollars; they agree to purchase cows, at the highest price per head that will allow each man to invest all his how money; many cows can each man purchase? Ans. A 5, B 10, and C 16.

MULTIPLES.

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

101. 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.

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

What is a multiple? A common multiple? The least common multiple?

103. From the definition (100) 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.

104. To find the least common multiple.

FIRST METHOD.

From the nature of prime numbers we derive the following principles:

I. If a number exactly contain another, it will contain all the prime factors of that number.

II. If a number exactly contain two or more numbers, it will also contain all the prime factors of those numbers.

III. The least number that will exactly contain all the prime factors of two or more numbers, is the least common multiple of those numbers.

1. Find the least common multiple of 30, 42, 66, and 78.

2× 3 × 13 × 11 × 7 × 5 = 30030, Ans.78, viz. :

X

2 X 3 X 13.

We here have all the prime factors of 78, and also all the factors of 66, except the factor 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 42 except the factor 7. Annexing 7 to the scries of factors, 2 X 3 X 13 X 11 X 7,

and we have all the prime factors of 78, 66, and 42, and also all the

How can a common multiple of two or more numbers be found? First principle derived from prime numbers? Second? Third?

Give analysis.

factors of 30 except the factor 5. Annexing 5 to the series of factors, 2 X 3 X 13 X 11 X 7 X 5,

and we have all the prime factors of each of the given numbers; and hence the product of the series of factors is a common multiple of the given numbers, (II.) And as no factor of this series can be omitted without omitting a factor of one of the given numbers, the product of the series 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. Take 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 used as many times, as a factor of the multiple, as the greatest number of times it appears in any of the given numbers.

EXAMPLES FOR PRACTICE.

2. Find the least common multiple of 7, 35, and 98. Ans. 490.

3. Find the least common multiple of 24, 42, and 17. Ans. 2856.

4. What is the least common multiple of 4, 9, 6, 8?

Ans. 72.

5. What is the least common multiple of 8, 15, 77, 385? Ans. 9240.

6. What is the least common multiple of 10, 45, 75, 90? Ans. 450.

7. What is the least common multiple of 12, 15, 18, 35? Ans. 1260.

Rule, first step? Second? What caution is given ›

4*

105. and 12?

SECOND METHOD.

1. What is the least common multiple of 4, 6, 9,

OPERATION.

4

6

9

2

2

3

3

9..

3..9.. 3

12

6

2 X2 X3 X 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 multiple sought. Dividing as many of the numbers as are divisible by 2, we write the quotients and the undivided number, 9, in a line underneath. We now perceive that some of the numbers in the second line contain the factor 2; hence the least common multiple must contain another 2, and we again divide by 2, omitting to write down 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, (104, III.)

2. What is the least common multiple of 10, 12, 15, and 75?

2,

OPERATION.

5 10..
.. 12..15..75

ANALYSIS. We readily see that 2 and 5 are among the factors of the given numbers, and must be factors of the least common multiple; hence we divide every number that is divisible by either of these factors or by their product; thus, we divide 10 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 afterwards 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.

2 X 5 X2 X3 X 5300, Ans.

3. What is the least common multiple of 6, 15, 35, 42, and 70?

Give explanation.

70; and whatever will contain 42 and 70 must contain 6 and 35. Hence we have only to find the least common multiple of the remaining numbers, 15, 42, and 70.

From these examples we derive the following

RULE. I. Write the numbers in a line, omitting any of the smaller numbers that 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 least common multiple required.

EXAMPLES FOR PRACTICE.

4. What is the least common multiple of 12, 15, 42, and 60? Ans. 420.

5. What is the least common multiple of 21, 35, and 42? Ans. 210.

6. What is the least common multiple of 25, 60, 100, and 125? Ans. 1500. 7. What is the least common multiple of 16, 40, 96, and 105? Ans. 3360.

8. What is the least common multiple of 4, 16, 20, 48, 60, and 72? Ans. 720. 9. What is the least common multiple of 84, 100, 224, and Ans. 16800.

300?

Rule, first step? Second? Third ?