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the division by canceling the 7, which is a factor both of the dividend and the divisor. (Arts. 88, 90.)

Operation. 7)63X4X7 The product of 63 X 4, the other factors of 252 Ans.

the dividend, is the quotient.

26. In 32 times 84, how many times 8? Ans. 336. 27. In 35 times 95, how many times 7? 28. In 48 times 133, how

many

times 8? 29. In 96 times 156, how many times 12 ?

30. Divide 168 x2x7 by 7 x 3. Operation.

We cancel the factor 7, which is comXX3)168X2XX

mon to the divisor and dividend, then 3)336

divide the product of 168 into 2 by 3. 112 Ans. 31. Divide the product of 8, 6, and 12, by the product of 2, 6, and 8.

Solution. &X6X8)8X6X12=6 Ans.

Note. We cancel the factors 2 and 6 in the divisor and the 12 in the dividend, &c. Canceling the same factor both in the divisor and dividend, is in effect dividing them both by the same number, and consequently does not effect the quotient. (Art. 88.) Hence,

91. a. When the divisor and dividend have factors common to both, the division may be performed by canceling the common factors, and then dividing those that are left as before.

32. Divide the product of 7, 9, 15, and 8 by the product of 5, 7, and 8.

33. Divide the product of 6, 3, 7, and 4 by the product of 12 and 6.

34. Divide the product of 2, 28, and 15 by 30. 35. . Divide the product of 5, 6, and 56 by 7x8.

Note. This method of contracting arithmetical operations, is called Cancelation. It applies with great advantage to that class of examples and problems which involve both multiplication and division ; that is, when the product of two or more numbers is to be divided by another number, or by the product of two or more numbers. Its further developments and application may be seen in reduction of compound fractions to simple ones ; in multiplication and division of fractions; in simple and compound proportion, &c., &c.

GREATEST COMMON DIVISOR.

92. A Common Divisor of two or more numbers, is a number which will divide them without a remainder. Thus 2 is a common divisor of 4, 6, 8, 12, 16, &c.

93. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide them without a remainder. Thus 6 is the greatest common divisor of 12, 18, and 24.

Obs. 1. One number is said to be a measure of another, when the former is contained in the latter any number of times without a remainder. Hence, a Com. divisor is often called a Common Measure.

2. It will be seen that a common divisor of two or more numbers, is simply a factor which is common to those numbers, and the greatest common divisor is the greatest factor common to them. Hence,

94. To find a common divisor of two or more numbers.

Resolve each number into two or more factors, one of which shall be common to all the given numbers.

Ex. 1. Find a common divisor of 8, 10, and 12.

Solution. 8 may be resolved into the factors 2 and 4; that is, 8=2X4; 10=2x5; and 12=2x6. The fac tor 2 is common to each number and is therefore a common divisor of them.

2. Find a common divisor of 9, 15, 18, and 24.

Obs. The following facts may assist the learner in finding common divisors :

1. Any number ending in 0, or an even number, as 2, 4, 6, &c. may be divided by 2.

2. Any number ending in 5 or 0, may be divided by 5. 3. Any number ending in 0, may be divided by 10.

4. When the two right hand figures are divisible by 4, the whole number may be divided by 4.

3. Find a common divisor of 16, 20, and 36.

Quest.-92. What is a common divisor of two or more numbers ? 93. What is the greatest common divisor of two or more numbers ? Obs. When is one number said to be a measure of another? What is a common divisor sometimes called ? 94. How do you find a common di. visor of two or more numbers ?

4. Find a common divisor of 35, 50, 75, and 80.
5. Find a common divisor of 148 and 184.
6. Find a common divisor of 126 and 4653.

95. No two numbers which have not a common factor, can have a common divisor greater than a unit. Thus the factors of 8 are 2 and 4; the factors of 15 are 3 and 5; hence, 8 and 15 have no common divisor.

96. To find the greatest common divisor of two numbers.

Divide the greater number by the less; then the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor.

7. What is the greatest common divisor of 70 and 84 ? Operation.

Dividing 84 by 70, the remainder is 14; 70)84(1 then dividing 70 (the preceding divisor)

70 by 14, (the last remainder,) nothing re14)7005

mains. Hence, 14, the last divisor, is the 70 greatest common divisor.

8. What is the greatest common divisor of 63 and 147 ? 9. What is the greatest common divisor of 91 and 117 ?

10. What is the greatest common divisor of 247 and 323?

11. What is the greatest common divisor of 285 and 465 ?

12. What is the greatest common divisor of 2145 and 3471?

97. To find the greatest common divisor of more than two numbers.

First find the greatest common divisor of any two of them, then, that of the common divisor thus obtained and of another given number, and so on through all the given

QUEST.-95. If two numbers have not a common factor, what is true as to a common divisor? 96. How find the greatest common divisor of two numbers ? 97. Of more than two ?

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 nuinbers 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.

Analysis.-6=2X3; and 10=2 X 5. It is evident that the number required must contain all the different factors which are in each of the given nurnbers ; otherwise it will not be a common multiple of them. (Art. 99. Obs.) The continued product of the factors 2 x3x2x 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 X3 X5=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.

We divide both numbers by 2. This re

solves them into factors, and the divisor 2)6 10

and quotients contain all the different fac3

5

tors found in each of the given numbers 2 X3 X5=30

once and only once. Then we multiply the divisor and quotients together and the product is 30, the least common multiple required. Hence,

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?

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