is any number that will divide them without a remainder; thus, 2 is a common divisor of 2, 4, 6, and 8.

ART. 122. To find a common divisor of two or more numbers.

Ex. 1. What is the common divisor of 10, 15, and 25 ?

OPERATION.

10=5X2

Ans. 5.

We resolve each of the given numbers into two factors, one of which is common to all of them. 15=5X3 In the operation 5 is the common factor, and there25 = 5 × 5 fore must be a common divisor of the numbers. RULE. Resolve each of the given numbers into two factors, one of which is common to all of them, and this common factor is a common divisor.

NOTE. A number is divisible by 2, when the last figure is even; by 4, when the last two figures are divisible by 4; by 8, when the last three figures are divisible by 8; by 5, when the last figure is 5; and by 5 or 10, when the last figure is 0.

EXAMPLES FOR PRACTICE.

2. What is the common divisor of 3, 9, 18, 24? Ans. 3. 3. What is the common divisor of 4, 12, 16, 28?

Ans. 2 or 4.

4. What is the common divisor of 12, 16, 32, 48?

Ans. 4.

5. What is the common divisor of 14, 21, 35,

49?

Ans. 7.

6. What is the common divisor of 8, 16, 24, 48, 64, 96, 128, and 144?

THE GREATEST COMMON DIVISOR.

Ans. 4.

ART. 123. The greatest common divisor of two or more numbers is the greatest number that will divide each of them without a remainder.

ART. 124. To find the greatest common divisor of two or more numbers.

Ex. 1. What is the greatest common divisor or measure of 84 and 132? Ans. 12.

QUESTIONS.

Art. 122. What is the rule for finding a common divisor.? When is a number divisible by two? By 4? By 8? By 5? By 10?Art. 123. What is the greatest common divisor of two or more numbers ?

OPERATION.

84) 132 (1

84

48) 84 (1
48

36) 48 (1
36
12) 36 (3
36

As 12 will divide 36, it is evident it will also divide 48, which is equal to 1236. It will also divide 84; because 84 is equal to 36+ 48; for, as 12 will divide each of these numbers, it is evident it will divide their sum. For the same reason, it will also divide 132, which is equal to 84 +48. We therefore find, that 12 is a common divisor of 84 and 132. To prove that 12 is the greatest common divisor, we resolve 84 and 132 into their prime factors; thus, 84 = 2 × 2X 3X7, and 132=2x 2x3x 11. Now it is evident that 84 cannot be divided by any number except by one of its prime factors or the product of two or more of them. The same is true of 132. Both these numbers, therefore, can be divided by 12, since it is the product of the first three prime factors of each of them; thus, 2 × 2 × 3 = 12. Again, if any number greater than 12 will divide both of these numbers, it must be a number common to the factors 7 and 11; but 7 and 11 are prime to each other, and therefore can have no common factor greater than 1. Hence 12 is the greatest common divisor of 84 and 132.

RULE. 1. Divide the greater number by the less, and if there is a remainder, divide the last divisor by it, and so continue dividing the last divisor by the last remainder until nothing remains, and the last divisor is the greatest common divisor.

2. If there are more than two numbers, first find the greatest common divisor of two of them, and then of that common divisor and one of the other numbers, and thus proceed until all the numbers are used. The last common divisor, thus obtained, is the greatest common divisor required.

NOTE. From the preceding demonstration it may be seen, that the greatest common divisor can also be found by resolving the given numbers into their prime factors, and multiplying together those which are common to all the numbers; thus 2, 2, 3, are factors, common to 84 and 132, and their product, 12, is their greatest common divisor.

EXAMPLES FOR PRACTICE.

2. What is the greatest common divisor of 85 and 95 ?

Ans. 5.

3. What is the greatest common divisor of 72 and 168?

Ans. 24.

4. What is the greatest common divisor of 119 and 121?

Ans. 1.

QUESTIONS. Art. 124. How does it appear that 12 in the example is the greatest common divisor of 84 and 132? What is the rule for finding the greatest common divisor? What other mode of finding the greatest common divisor?

5. What is the largest number that will divide 324 and 586? Ans. 2.

6. What is the largest number that will divide 582 and 684 ?

7. What is the greatest common divisor of 32

8. What is the largest number that will divide 84

9. What is the greatest common divisor of 16,

10. What is the greatest common divisor of and 30?

Ans. 6. and 172 ? Ans. 4. and 1728 ? Ans. 12. 20, and 26 ? Ans. 2. 12, 18, 24, Ans. 6.

A COMMON MULTIPLE.

ART. 125. A multiple of a number is a number that can be divided by it without a remainder; thus 6 is a multiple of 3.

ART. 126. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus 12 is a common multiple of 3 and 4.

ART. 127. The least common multiple of two or more numbers, is the least number that can be divided by each of them without a remainder.

ART. 128. To find the least common multiple.

Ex. 1. What is the least common multiple of 6, 9, 12?

OPERATION.

316 9 12

22 3

4

1 3

2

Ans. 36.

In this operation we first divide the numbers by 3, a number that will divide most of them without a remainder, and write the quotients in a line below. We next divide by 2, writing down the quotients and undivided number, as before. Then, since these numbers are prime to each other, we multiply together the divisors, last quotients, and undivided number, which are all the prime factors of 6, 9, and 12, and thus obtain 36 for the least common multiple.

3 × 2 × 3 × 2 = 36

To prove that 36 is the least common multiple, we resolve 6,

QUESTIONS.-Art. 125. What is a multiple of a number? Art. 126. What is a common multiple of two or more numbers? - Art. 127. What is the least common multiple of two or more numbers? Art. 128. How does it appear that 36 is the least common multiple of 6, 9, and 12 ?

=

9, and 12 into their prime factors; thus, 6 2 X 3; 9: 3 X 3; 12 = 2 × 2 × 3. Now, since the least common multiple of two or more numbers must contain all the factors of those numbers, it is evident that, if any different prime factor occurs more than once in any one of the numbers, it must be taken the same number of times as a factor of the multiple, or it will not contain all the factors of the numbers.

In the example, 2 and 3 are the different prime factors, and as 3 occurs twice in 9, and 2 twice in 12, each must be taken twice as factors of the least common multiple; thus, 2 × 2 × 3 × 3: same as in the operation.

=

36, the

RULE. Divide by such a number as will divide MOST of the given numbers without a remainder, and set the several quotients with the several undivided numbers in a line beneath; and so continue to divide, until no number greater than unity will divide two or more of them. Then multiply all the divisors, the last quotients, and undivided numbers together, and the product is the least common multiple.

NOTE 1. Care must be taken always to divide by a number that will divide most of the given numbers, or a multiple may be obtained which is not the least common multiple.

NOTE 2. When one or more of the given numbers are factors of any one of the other numbers, the factor or factors may be cancelled, and a common multiple of the remaining numbers found as in other examples. Thus, if the common multiple of 5, 15, 30, 7, 14, and 28 were required, we might cancel the 5, 15, 7, and 14, because 5 and 15 are factors of 30, and 7 and 14 are factors of 28.

EXAMPLES FOR PRACTICE.

2. What is the least common multiple of 8, 4, 3, and 6 ?

Ans. 24.

3. What is the least common multiple of 7, 14, 21, and 15?

4. What is the least common multiple of and 8 ?

Ans. 240.

5. What is the least number that 10, 12, 16, 20, and 24 will divide without a remainder? 6. What is the least common multiple of 9, 8, 12, 18, 24, 36, and 72 ?

Ans. 72.

7. Five men start from the same place to go round a certain island. The first can go round it in 10 days; the second in 12 days; the third in 16 days; the fourth in 18 days; the fifth in 20 days. In what time will they all meet at the place from which they started ? Ans. 720 days.

QUESTIONS.-What is the rule for finding the least common multiple? What caution is given in the note?

§ XIX. FRACTIONS.

ART. 129. A FRACTION is an expression denoting a part of any number or thing.

The term fraction is derived from the Latin word frango, which signifies to break; from the idea that a number or thing is broken or separated into parts.

Fractions are of two kinds, Vulgar and Decimal.

VULGAR FRACTIONS.

ART. 130. A VULGAR* FRACTION is any part of an integer or whole number, expressed by two numbers one above the other with a line between them.

The number below the line is called the denominator, and the number above, the numerator;

The denominator shows into how many parts the whole number is divided, and gives a name to the fraction. The numerator shows how many of these parts are taken, or expressed by the fraction.

The numerator and denominator together are called the terms of the fraction.

ART. 131. There are six kinds of vulgar fractions, viz. proper, improper, mixed, simple or single, compound, and complex.

A proper fraction is one whose numerator is less than the denominator; as, .

An improper fraction is one whose numerator is equal to, or greater than, the denominator; as, §, f.

*The word vulgar here means common, and is employed in this connection to denote the kind of fractions in most common use.

QUESTIONS. Art. 129. What is a fraction? From what is the term derived, and what does it signify? How many kinds of fractions, and what are they called? - Art. 130. What is a vulgar fraction, and how expressed? What is the number above the line called? The number below the line? What does the denominator show? What the numerator ? What are the numerator and denominator together called? - Art. 131. How many kinds of vulgar fractions, and what are their names? Give the definition of each.