ART. 119. To find the least common multiple of two or more numbers.

FIRST METHOD.

One number is divisible by another, when it contains all the prime factors of that number.

Thus, 30 is divisible by 6, because 30-2×3×5, and 6=2X3; the prime factors of 6, which are 2 and 3, being also factors of 30.

One number is not divisible by another, unless it con, tains all the prime factors of that other.

Thus, 10 is not divisible by 6, because 3, one of the prime factors of 6, is not a factor of 10.

Hence, a common multiple of two or more numbers must contain all the prime factors in those numbers; and,

To be the least common multiple, (L. C. M.), it must not contain any prime factor not found in some one of the numbers. L. C. M. should be read, least common multiple. 1. What is the L. C. M. of 6 and 10?

SOLUTION.-By factoring, 6=2 × 3, and 10 2 X 5. A number composed of the factors 2, 3, and 5, will contain all the factors in each of the numbers 6 and 10, and will contain no other factor; therefore, cross out (cancel) the factor 2, in one of the numbers; the factors, 2 X3 X5 = 30, will be the L.

2. What is the L. C. M. of 6,

SOLUTION. By factoring the numbers, the prime factor 2 occurs once in 6, three times in 8, and twice in 12; hence it must

occur three times, and only three

OPERATION.

6=2×3 10=2X5

2X 3X5=30 Ans.

product of the remaining C. M. of 6 and 10. 8, and 12?

OPERATION.

6=2×3

8:

2 × 2 × 2

12

=

2 × 2 × 3

times, in the L. C. M.; therefore, 3× 2 × 2 × 2 = 24 Ans.

after reserving it as a factor

three times, cancel it in the other numbers.

The prime factor 3 occurs once in 6 and once in 12; hence, it must occur once, and only once, in the L. C. M. it once as a factor, cancel the other factor 3. then found by multiplying together the figures

After reserving The L. C. M. is not canceled.

Rule I.-Separate the numbers into prime factors; then multiply together ONLY such of those factors, as are necessary to form a product that will contain all the prime factors in each number, using no factor oftener than it occurs in any one number.

NOTE. The solution of Ex. 2, shows that the same factor must be taken the greatest number of times it occurs in either number. After factoring, cancel (cross out) the needless factors.

FIND THE LEAST COMMON MULTIPLE OF

5. 6, 8, 9.

Ans. 72. 8.

6. 6, 15, 35.

Ans. 210. 9.

9, 15, 18, 24. Ans. 360. 8, 15, 12, 30. Ans. 120.

7. 10, 12, 15.

Ans. 60. 10. 14, 21, 30, 35. Ans. 210.

SECOND METHOD.

ART. 120. The L. C. M. of two or more numbers, contains all the prime factors of each of the numbers once, and no other factors.

For, if it did not contain all the prime factors of any number, it would not be divisible by that number; and, if it contained any prime factor not found in either of the numbers, it would not be the least common multiple.

Thus, the L. C. M. of 4 (2×2), and 6 (2×3), must contain the factors 2, 2, 3, and no others.

1. Find the L. C. M. of 6, 9, and 12.

SOLUTION.-Arranging the num

bers as in the margin, we find that 2 is a prime factor common to two of them.

Hence, 2 must be a factor of the L. C. M.; therefore, place it on the left, and cancel it in the

OPERATION.

2)6 9 12

3)3 9 6
1 3

2 × 3 × 3 × 2 = 36 Ans.

numbers of which it is a factor, by dividing by it.

Next, observe that 3 is a factor common to the quotients and the remaining number, and hence, (Art. 111,) is a factor of the given numbers, and must be a factor of the L. C. M.;

therefore, place it on the left, and cancel it in each of the numbers in the 2d line, by dividing by it. As the numbers 3 and 2, in the 3d line, have no common factors to cancel, we do not divide them.

Thus we find, that 2, 3, 3, and 2, are all the prime factors in the given numbers; hence, their product, 2X3X3X2=36, is the L. C. M. of 6, 9, and 12.

In this operation, let the learner notice,

1st. The number 36 is a common multiple, because it contains all the prime factors in each of the numbers; it is the least C. M., because all the needless factors were canceled by dividing.

2d. To cancel needless factors, divide by a prime number. By dividing by a composite number, in some cases, all the needless factors are not canceled; thus, in the preceding example, 6 will exactly divide two of the numbers; but,

In dividing by 6, a factor, 3, is left uncanceled in the multiple 9, and thus the L. C. M. is not obtained.

2. Find the L. C. M. of 6 and 10.

Ans. 30.

Ans. 105.

Rule II.-1. Place the numbers in a line, divide by any prime number that will divide two or more of them without a remainder, and place the quotients and undivided numbers in a line beneath.

2. Divide this line as before: continue to divide till no number greater than 1 will exactly divide two or more of the numbers.

3. Multiply together the divisors and the numbers in the lowest line, and their product will be the least common multiple.

REM. If the given numbers contain no common factor, their product will be the L. C. M. Thus, the L. C. M. of 4, 5, and 9, is 4X5X9=180.

REVIEW.-119. When is one number divisible by another? Give an example. When not divisible? Give an example. What factors must the Com. Mul. contain?

119. What prime factors must the L. C. M. not contain? Find the L. C. M. of 6, 10, and 18, and explain the operation. What is Rule I? 119. NOTE. How often must the same factor be found in the L. C. M.?

XI. COMMON FRACTIONS.

ART. 121. A single thing, (Art. 1), is called a unit, or one, which may be divided into equal parts.

Thus, suppose 3 apples are to be equally divided between 2 boys: after giving one to each, there would remain one to be divided into two equal parts, to complete the division.

The equal parts into which a unit is divided are fractions. ART. 122. When a unit, or single thing, is divided into two equal parts, one of the parts is one-half.

If it is divided into three equal parts, one of the parts is one-third; two of the parts, two-thirds.

If divided into four equal parts, one of the parts is one-fourth; two of the parts, two-fourths; and three of the parts, three-fourths.

If divided into five equal parts, the parts are fifths; if into six equal parts, sixths, and so on. Hence,

When a unit is divided into equal parts, the parts are named from the number of parts into which the unit is divided.

REVIEW.-120. Ex. 1. Why divide by 2? By 3? Why multiply together the numbers 2, 3, 3, and 2? Why is 36 a Com. Mul. of 6, 9, and 12? Why the least? To cancel needless factors, why not divide by a composite number? What is Rule II?

120. REM. If the numbers contain no common factor, how is their L. C. M. found? 121. How do you divide 3 apples equally between 2 boys? When a unit is divided into equal parts, what are the parts called?

ART. 123. The value of one of the parts depends on the number of parts into which the unit is divided.

Thus, if 3 appies of equal size be divided, one into 2, another into 3, and another into 4 equal parts, the thirds will be less than the halves, the fourths less than the thirds.

ART. 124. Fractions are divided into two classes, Common and Decimal.

Common Fractions are expressed by two numbers, one above the other, with a horizontal line between them.

Thus, one-half is expressed by ; two-thirds by .

The number below the line is the denominator: it denominates, or gives name to the fraction. It shows the number of parts into which the unit is divided.

The number above the line is the numerator: it numbers the parts, showing how many parts are taken.

Thus, in the fraction, the denominator, 5, shows that the unit is divided into five equal parts, and the numerator, 3, shows that the fraction contains 3 of those parts.

The numerator and denominator together, are called the terms of the fraction. Thus, the terms of 3, are 3 and 5.

ART. 125. ANOTHER METHOD.

In the definition of numerator and denominator, reference is had to a unit only. This is the simplest method of considering a fraction; but, there is another mode:

ILLUSTRATION. To divide 2 apples equally among 3 boys, divide each apple into three equal parts, making 6 parts in all; then give to each boy 2 of the parts, expressed by 3.

REVIEW.-122. When a unit is divided into two equal parts, what is one part called? When divided into three equal parts, what is one part called? What two parts? When divided into four equal parts, what is one part called? Two parts? Three parts? When a unit is divided into equal parts, from what are the parts named ?

123. On what does the value of one of the parts depend? Which is greater, 1-half or 1-third? 1-third or 1-fourth? 1-fourth or 1-fifth?

124. Into what two classes are fractions divided? How are common fractions expressed? What is the number below the line? Why? What the number above the line? Why? What are the terms?