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The divisor sought being a common divisor of 227 and 908, it must, by the Second Principle, be a divisor of their sum, 1135. The divisor sought being a common divisor of 908 and 1135, it must, by the Second Principle, be a divisor of their sum, 2043.

The divisor sought being a divisor of 2043, it must, by the First Principle, be a divisor of any multiple of 2043, that is, of 4086.

And being a divisor of 2043 and 4086, it must, by the Second Principle, be a divisor of their sum, 5221.

Finally. Since no number can have a divisor greater than itself; and since the divisor sought must be a divisor of 227, 2043, and 5221, it follows that the greatest common divisor of 2043 and 5221 is 227.

2. Find the Greatest Common Divisor of 2760 and 7245. Ans. 345.

3. Find the Greatest Common Divisor of 2387 and 3255. Ans. 217. 4. Find the Greatest Common Divisor of 8652 and 20909. Ans. 721. 5. Find the Greatest Common Divisor of 768, 1408, 3200.

Suggestion. - Find the Greatest Common Divisor of two of the given numbers. Then find the Greatest Common Divisor of it and the other number.

LEAST COMMON MULTIPLE.

91. A multiple of any number is a number which will exactly contain it. Thus, 24, 36, 48, 72, 84, and 96, are multiples of 12.

92. A Common Multiple of two or more numbers is a number which is exactly divisible by each of them. Thus, 144 is a common multiple of 72, 48, 24, 16, 12, 9, 8, 6, 4, 3, 2.

93. The Least Common Multiple of two or more numbers is the least number that will contain each of them an integral number of times. Thus, 60 is the least common multiple of 30, 20, 15, 12, 10, 5, 4, 3, 2.

FUNDAMENTAL PRINCIPLES.

The Least Common Multiple of several numbers is the product of all the different prime factors of those numbers, taken the greatest number of times they are found in either of the given numbers.

94. a. The Least Common Multiple must contain all the different prime factors found in the numbers themselves, otherwise it would not be divisible by each of them.

b. The Common Multiple must not contain any prime factor a greater number of times than it is found in any of the numbers, otherwise it will not be the Least Common Multiple.

FIRST METHOD.

95. To find the Least Common Multiple by factoring.

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Example 1. Find the Least Common Multiple of 48, 64, 72, 96.

Solution. The prime factors of 48 are 2, 2, 2, 2, and 3.

The prime factors of 64 are 2, 2, 2, 2, 2, and 2.

The prime factors of 72 are 2, 2, 2, 3, and 3.

The prime factors of 96 are 2, 2, 2, 2, 2, and 3.

Here, the several different factors are 2 and 3.

Since 2 occurs six times as a factor in 64, it must occur six times as a factor in the Least Common Multiple.

Since 3 occurs twice as a factor in 72, it must occur twice in the

Least Common Multiple. Hence, the Least Common Multiple of 48, 64, 72, and 96 is 576.

2. Find the Least Common Multiple of 12, 14, 18, and 21. Ans. 252.

3. Find the Least Common Multiple of 12, 21, 20, and 48.

4. Find the Least Common Multiple of 11, 13, 26, and 99.

5. Find the Least Common Multiple of 13, 39, 56, and 63.

6. Find the Least Common Multiple of 8, 10, 12, 18, and 25.

7. Find the Least Common Multiple of 4, 5, 6, 7, and 8.

8. Find the Least Common Multiple of 16, 24, 36, and 72.

9. Find the Least Common Multiple of 18, 32, 42, and 56.

10. Find the Least Common Multiple of 20, 34, 54, and 65.

96. The product of several numbers which are prime to each other, is their least common multiple.

97. Numbers are prime to each other when no whole number is an exact divisor of all of them.

SECOND METHOD.

98. This method differs from the first only in the manner of resolving the given numbers into their prime factors.

Example 1. - Find the Least Common Multiple of 18, 27, 36, 44, 56, and 63.

Suggestion. —Arrange the several numbers as below, and divide by any prime number that is an exact divisor of two or more of them; write the quotients beneath, and bring down the numbers undivided. Proceed in the division until no prime number will divide two of the given numbers.

The product of the several divisors, quotients, and remainders, will be the least common multiple.

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Here, 2 occurs three times as a factor; 3 occurs three times; 7 occurs once, and 11 once. Hence, 2 X2 X2 × 3 × 3 × 3 × 7×11, or 16632, is the least common multiple of 18, 27, 36, 44, 56, and 63.

2. Find the Least Common Multiple of 15, 25, 35, 45.

3. Find the Least Common Multiple of 14, 28, 42, 56, 64.

4. Find the Least Common Multiple of 10, 20, 30, 40, 50.

5. Find the Least Common Multiple of 16, 20, 24, 28, 32.

6. Find the Least Common Multiple of 18, 22, 32, 44, 54.

7. Find the Least Common Multiple of 22, 33, 44, 55, 66.

BUSINESS METHODS.

99. Business Calculations may often be greatly facilitated by using the following fractional parts of One Dollar.

One-third of one dollar equals 33 cents.
One-sixth of one dollar equals 163 cents.
One-eighth of one dollar equals 12 cents.
One-twelfth of one dollar equals 8 cents.
One-sixteenth of one dollar equals 64 cents.
One-half of one dollar equals 50 cents.
One-fourth of one dollar equals 25 cents.
One-fifth of one dollar equals 20 cents.
One-tenth of one dollar equals 10 cents.
One-twentieth of one dollar equals 5 cents.

APPLICATIONS.

1. Find the cost of 64 arithmetics, at 37 cents

apiece.

Explanation.—371⁄2 is three times 121.

12 cts. is of one dollar.

Then, 37 cents is of one dollar.

64 arithmetics, at $1, cost $64.

64 arithmetics, at 37

or $24.

cents, cost of $64,

2. Find the cost of 96 readers, at 87 cents apiece.

Explanation.-87 is seven times 12.

12 cents is of one dollar.

Then, 87 cents is of one dollar.

96 readers, at $1, cost $96.

96 readers, at 871 cents, cost of $96, or $84.

3. Find the cost of 105 bushels of wheat, at 662 cents per bushel.

Explanation.

-663 is twice 331.

33 cents is of one dollar.

Then, 663 cents is of one dollar.

105 bu. of wheat, at $1, cost $105.

105 bu. of wheat, at 663 cents, cost of $105, or $70.

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