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numbers. The last common divisor found, will be the one required.

13. What is the greatest common divisor of 63, 105, and 140 ?

Suggestion. Find the greatest common divisor of 63 and 105, which is 21. Then, that of 21 and 140.

Ans. 7.

14. What is the greatest common divisor of 16, 24, and 100?

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

LEAST COMMON MULTIPLE.

98. One number is said to be a multiple of another, when the former contains the latter a certain number of times without a remainder. Thus 4 is a multiple of 2; 10 is a multiple of 5, &c.

OBS.-A multiple is therefore a composite number, and the number thus contained in it, is always one of its factors.

99. A common multiple of two or more numbers, is a number which can be divided by each of them without a remainder.

Thus, 12 is a common multiple of 2, 3, and 4; 15 is a common multiple of 3 and 5, &c.

OBS. A common multiple is also a composite number of which each of the numbers contained in it, must be a factor taken once or more.

100. The continued product of two or more given numbers will always form a common multiple of those numbers.

The same numbers, therefore, may have an unlimited number of common multiples; for, multiplying their con

QUEST.-98. What is a multiple of a number? Obs. What kind of a number is a multiple? 99. What is a common multiple? Obs What kind of a number is a common multiple? 100. How may a common multiple of two or more numbers be obtained? How many common multiples may there be of any given numbers?

tinued product by any number, will form a new common multiple. (Art. 99. Obs.)

101. The least common multiple of two or more numbers, is the least number which can be divided by each of them without a remainder. Thus 12 is the least common multiple of 4 and 6, for it is the least number which can be exactly divided by them.

It is evident

Analysis.-6=2×3; and 10=2×5. that the number required must contain all the different factors which are in each of the given numbers; otherwise it will not be a common multiple of them. (Art. 99. Obs.) The continued product of the factors 2×3×2× 5=60, is exactly divisible by 6 and 10, but it will be easily perceived that it is twice as large as is necessary to be a common multiple of them. We also perceive that the factor 2 is common to both the given numbers; hence it is that the continued product is twice too large. If, therefore, we retain this factor only once, the continued product of the rest 2 ×3×5=30, and is the smallest number that can be found exactly divisible by 6 and 10, and is therefore the least common multiple of them.

Operation. 2)6 10

"

We divide both numbers by 2. This resolves them into factors, and the divisor and quotients contain all the different factors found in each of the given numbers once and only once. Then we multiply the divisor and quotients together and the product is 30, the least common multiple required. Hence,

3 "1 5 2x3x5=30

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

Write the given numbers in a line with two points between them. Divide by the smallest number which will divide any two or more of them without a remainder, and set

QUEST.-101. What is the least common multiple of two or more numbers? 102. How is the least common multiple of two or more numbers found?

the quotients and the numbers not divided in a line below. Divide this line and set down the results as before; thus continue the operation till there are no two numbers which can be divided by any number greater than 1. The continued

product of the divisors into the numbers in the last line, will be the least common multiple required.

16. Find the least common multiple of 6, 8, and 12.

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OBS. 1. In the first operation, we divide by the smallest numbers which will divide any two of the given numbers without a remainder, and the product of the divisors, &c., is 24, which is the answer required.

In the second operation, we divide by 6, then by 2, which are not the smallest numbers that will exactly divide two of the given numbers, and the continued product of the divisors into the figures in the last line is 48, which is not the least common multiple. Hence,

2. We must divide, in all cases, by the smallest number that will divide any two of the given numbers exactly; otherwise, the divisor may contain a factor common to it and some one of the quotients, or undivided numbers in the last line, and consequently the continued product of them will be too large for the least common multiple.

17. Find the least common multiple of 4, 9, and 12. 18. Find the least common multiple of 16, 12, and 24. 19. Find the least common multiple of 15, 9, 6, and 5. 20. Find the least common multiple of 10, 6, 18, 15. 21. Find the least common multiple of 24, 16, 15, 20. 22. Find the least common multiple of 25, 60, 72, 35. 23. Find the least common multiple of 63, 12, 84, 72. 24. Find the least common multiple of 54, 81, 14, 63. 25. Find the least common multiple of 12, 72, 36, 144.

QUEST.-Obs. Why do you divide by the smallest number that will divide two or more without a remainder.

SECTION VI.

FRACTIONS.

MENTAL EXERCISES.

ART. 103. When a number or thing is divided into two equal parts, one of those parts is called one half. If the number or thing is divided into three equal parts, one of the parts is called one third; if it is divided into four equal parts, one of the parts is called one fourth, or one quarter; two of the parts, two fourths; three, three fourths; if divided into five equal parts, the parts are called fifths; if into six equal parts, sixths; if into ten, tenths; if into a hundred, hundredths, &c. That is,

When a number or thing is divided into equal parts, the parts always take their name from the number of parts into which the thing or number is divided.

104. The value of one of these equal parts manifestly depends upon the number of parts into which the given number or thing is divided. Thus, if an orange is successively divided into 2, 3, 4, 5, 6, &c., equal parts, the thirds will be less than the halves; the fourths, than the thirds; the fifths, than the fourths, &c.

Ex. 1. What is one half of 2 cents? Of 4 cents? 6? 8? 16? 18? 20? 24? 30? 40? 50? 60? 70? 80? 100 ? 2. What is one third of 6 cents? Of 9? 12? 15?

QUEST.-103. What is meant by one half? How many halves make a whole one? What is meant by one third? How many thirds make a whole one? What is meant by a fourth? 3 fourths? What are fourths sometimes called? How many fourths make a whole one? What is meant by fifths? By sixths? Eighths? How many sevenths make a whole one? How many tenths? What is meant by twentieths? By hundredths? When a number or thing is divided into equal parts, from what do the parts take their name? 104. Upon what does the value of one of these equal parts depend? Which is the greater, a half or a third? A sixth or a fourth? A seventh or a tenth?

OBS. A half of any number, it will be perceived, is equal to as many units, as 2 is contained times in that number; a third of a number is equal to as many units, as 3 is contained times in the given number; a fourth is equal to as many, as 4 is contained in it, &c.

3. What is a third of 12? Of 15? 18? 21? 24? 27? 30? 36? 39? 45? 60?

4. What is a fourth of 8 dollars? Of 12? 16? 20? 24? 28? 32? 36? 40? 44? 48?

5. What is a fifth of 5? 10? 15? 20? 25? 30? 35? 40? 45? 50? 55? 60? 100 ?

6. What is a sixth of 12? 18? 24? 36? 30? 48? 60? 54 42? 72?

7. What is a seventh of 14? 28? 35? 21? 42? 56 ? 49? 63?

8. What is an eighth of 16? 24? 40? 32? 64? 48? 56 72 88?

9. What is a ninth of 9? 18? 36? 27? 45? 54? 72 ? 63 81? 99 ?

10. What is a tenth of 20? 40? 60? 50? 30? 100? 90? 120?

11. What part of 2 is 1?

Ans. One half.

12. What part of 3 is 1? Of 4? 5? 7? 10? 15? 19? 37? 200?

13. What part of 3 is 2?

Suggestion. Two is two times the third part of 3, or two thirds of 3.

14. What part of 5 is 2? is 3? is 4? is 5? is 6? is 8? is 9?

15. What part of 8 is 3? is 7? is 6? is 9? is 8? 12? 15?

16. What part of 17 is 5? 8? 9? 13? 15? 16? 20? 17. What part of 100 is 13? 29? 63? 75? 92? 18. If 1 half an orange cost 2 cents, what will a whole orange cost?

Suggestion. If 1 half of an orange cost 2 cents, 2 halves, or a whole orange, will cost twice as much; and 2 times 2 cents are 4 cents. Ans. 4 cents.

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