common divisor cannot be greater than the difference between the two numbers, which, in this case is 18. Therefore, we have

PROPOSITION VII. The greatest common divisor of two numbers is obtained by dividing the greater by the less, then dividing the divisor by the remainder, and continuing to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought.

Q. Will the common divisor of two numbers divide their remainder after division? How do you find the greatest common divisor of two numbers?

3. Find the greatest common divisor of the two numbers 63 and 81.

4. Find the greatest common divisor of 315 and 405.

Ans. 45.

Ans.

5. What is the greatest common divisor of the two numbers 2205 and 2835? 6. Find the greatest common divisor of 1157 and 623.

Ans.

7. Find the greatest common divisor of 792 and 1386. Ans. 198.

NOTE-If it be required to find the greatest common divisor of more than two numbers, find first the greatest common divisor of two of them, then of that common divisor and one of the remaining numbers, and so on, for all the numbers the last common divisor will be the greatest common divisor of all the numbers.

8. What is the greatest common divisor of 246, 372,

and 522 ?

Ans.

9. What is the greatest common divisor of 492, 744 and 1044?

Ans. 12,

LEAST COMMON MULTIPLE.

§ 87. A number is said to be a common multiple of two or more numbers, when it can be divided by each of them without a remainder. For example, 6 is a common multiple of 2 and 3, because it is exactly divisible by each of them. So likewise, 12 is a common multiple of 2, 3, 4, and 6, because it is divisible by each of them.

The least common multiple of two or more numbers, is the least number which they will separately divide without a remainder. For example, 12 is a common multiple of 2 and 3, but it is not the least common multiple, since 6 is also divisible by 2 and 3. Now 6 being the least number which is so divisible, it is the least common multiple of 2 and 3.

To find the least common multiple of several numbers, we have the following

RULE.

I. Place the numbers on the same line, and divide by the least number that will divide two or more of them without a remainder, and set down in a line below the quotients and the undivided numbers.

II. Divide as before, until there is no number greater than 1 that will exactly divide any two of the numbers: then multiply together the numbers of the lower line, and the divisors, and the product will be the least common multiple. If, in comparing the numbers together we find no common divitheir product is the least common multiple.

sor,

EXAMPLES.

1. Find the least common multiple of 3, 4 and 8.

We first see, that 2 will

divide 4 and 8, but as it will not divide 3, we bring down 3 into the 2nd line: we again see that 2 is a common divisor of 2 and 4; and as there is no com

mon divisor between any two of the numbers of the last line, it follows that 2×1×3 multiplied by the two divisors, is the least common multiple.

twelve-sixths?

In fifteen-sixths? In eighteen-sixths? In twenty-five sixths? In twenty-six sixths? In thirtyseven sixths?

40. How many whole units are there in seven-sevenths? In fourteen-sevenths? In nineteen-sevenths? In twentynine sevenths? In thirty-five sevenths? In forty-two

sevenths?

41. How many whole units are there in eight-eighths? In twenty-four eighths? In sixteen eighths? In thirty eighths? In thirty-four eighths? In forty eighths?

42. How many whole units are there in nine-ninths? In eighteen ninths? In twenty-four ninths? In thirtyfive-ninths? In forty-five ninths?

43. How many whole units are there in ten-tenths? In twenty tenths? In thirty tenths? In forty-five tenths? In sixty tenths?

44. How many whole units are there in eleven-elevenths? In twenty-two elevenths? In thirty-three elevenths? In forty-four elevenths? In sixty elevenths? In sixty-seven elevenths?

45. How many whole units are there in twelve-twelfths? In twenty-four twelfths? In twenty-six twelfths? In forty-eight twelfths? In fifty twelfths? In sixty twelfths?

46. What is the sum of one-half and one-fourth? What is the sum of one-half and three-fourths ? Of threefourths and six fourths?

47. What is the sum of one-third and one-sixth? (How many sixths is one-third equal to?) What is the sum of two-thirds and five-sixths? Of one-third and five sixths?

SECTION II.

§ 77. There are five kinds of Vulgar Fractions, Proper, Improper, Simple, Compound, and Mixed.

A PROPER FRACTION is one in which the numerator is less than the denominator. The value of every proper fraction is less than 1. (See § 44.)

The following are proper fractions:

AN IMPROPER FRACTION is one in which the numerator is equal to, or exceeds the denominator. Such fractions are called improper fractions because they are equal to, or exceed unity. When the numerator is equal to the denominator the value of the fraction is 1; in every other case the value of an improper fraction is greater than 1. The following are improper fractions:

14
2,5,』3,7,#,??,!;, !.

A SIMPLE FRACTION is a single expression. A simple fraction may be either proper or improper. The following are simple fractions:

A COMPOUND FRACTION is a fraction of a fraction, or several fractions connected together with the word of between them.

The following are compound fractions:

¦ of }, } of } of {}, ¦ of 3, ‡ of ¦ of 4.

A MIXED NUMBER is made up of a whole number and a fraction. The whole numbers are sometimes called integers. The following are mixed numbers:

31, 41, 63, 53, 65, 34,

Q. How many kinds of Vulgar Fractions are there? What are they? What is a proper fraction? Is its value greater or less than 1 ? What is an improper fraction? Why is it called improper? When is its value equal to 1? What is a simple fraction? What is a compound fraction? What is a mixed number? Give an example of a proper fraction? Of an improper fraction? Of a simple fraction? Of a compound fraction? Of a mixed fraction? Is four-ninths a proper or improper fraction? What kind of a fraction is six-thirds? What is its value? What kind of a fraction is nine-eighths? What

is its value? What kind of a fraction is one-half of a third? What kind of a fraction is two and one-sixth? Four and a seventh? Eight and a tenth?

$78. The numerator and denominator of a fraction, taken together, are called the terms of the fraction. Hence, every fraction has two terms.

Q. What are the terms of a fraction? What are the terms of the fraction three-fourths? Of five-eighths? Of six-sevenths?

§ 79. A whole number may be expressed fractionally by writing 1 below it for a denominator. Thus,

3 may be written and is read, 3 ones.

5

917

5
I

6

5 ones.

6 ones.

8 ones.

But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 8 ones to 8. Hence, the value of a number is not changed by placing 1 under it for a denominator.

Q. How may a whole number be expressed fractionally? Does this alter its value? Give an example?

§ 80. If an apple be divided into 6 equal parts,

and generally, the denominator shows into how many equal parts the unit is divided, and the numerator how many of the parts are taken.

Hence, also, we may conclude, that,

PROPOSITION I. If the numerator of a fraction be multiplied by any number,the denominator remaining unchanged, the value of the fraction will be increased as many times as there are units in the multiplier. Hence, to multiply a frac tion by a whole number, we simply multiply the numerator by the number.

Q. If an apple be divided in six equal parts how do you express one of those parts? Two of them? Three of them? Four of them? Five of them? Repeat the proposition? How do you multiply a fraction by a whole number?