168

203

2

84

3

70

2

14

know that the greatest common divisor, whatever it be, must divide 2 times 84, or 168, (I.) Then since it will divide both 168 and 203, it must divide their difference, 35, (III.) It will also divide 2 times 35, or 70, (I ;) and as it will divide both 70 and 84, it must divide their difference, 14, (III.) It I will also divide 2 times 14 or 28, (I;) and as it will divide both 28 and 35, it must divide their difference, 7, (III;) hence, it cannot be greater than 7.

35

28

Thus we have shown,

1st. That is a common divisor of the given numbers. 2d. That their greatest common divisor, whatever it be, Hence it must be 7.

cannot be greater than 7. From this example and

analysis, we derive the following

RULE. I. Draw two verticals, and write the two numbers, one on each side, the greater number one line above the less.

II. Divide the greater number by the less, writing the quotient between the verticals, the product under the dividend, and the remainder below.

III. Divide the less number by the remainder, the last divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor sought.

IV. If more than two numbers be given, first find the greatest common divisor of two of them, and then of this divisor and one of the remaining numbers, and so on to the last; the last common divisor found will be the greatest common divisor of all the given numbers.

1. When more than two numbers are given, it is better to begin with the least two. 2. If at any point in the operation a prime number occur as a remainder, it must be a common divisor, or the given numbers have no common divisor.

Rule, first step? Second? Third? Fourth? What relation have numbers when their difference is a prime number?

EXAMPLES FOR PRACTICE.

1. What is the greatest common divisor of 221 and 5512?

2. Find the greatest common divisor of 154 and 210.

Ans. 14.

3. What is the greatest common divisor of 316 and 664 ? Ans. 4.

4. What is the greatest common divisor of 679 and 1869? Ans. 7.

5. What is the greatest common divisor of 917 and 1495? Ans. 1.

6. What is the greatest common divisor of 1313 and 4108? Ans. 13.

7. What is the greatest common divisor of 1649 and 5423 ? Ans. 17.

The following examples may be solved by either of the foregoing methods.

8. John has 35 pennies, and Charles 50: how shall they arrange them in parcels, so that each boy shall have the same number in each parcel? Ans. 5 in each parcel.

9. A speculator has 3 fields, the first containing 18, the second 24, and the third 40 acres, which he wishes to divide into the largest possible lots having the same number of acres in each; how many acres in each lot? Ans. 2 acres.

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 money; how 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.

We here have all the prime factors of 78, and also all the factors of 6, except the factor 11. Annexing 11 to the series of factors,

2 × 3 × 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 series 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

Third ?

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

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?