divisor or dividend, or of both, have been canceled. When all the factors of both divisor and dividend are canceled the result is is=1 1.

EXAMPLE.

Divide the product of 7, 8, 11, and 36, by the product of 2, 4,

12, and 14.

OPERATION.

ANALYSIS.-Having first indicated the operations to be performed, we observe that 8 in the dividend is 16 Ans. equal to 2 × 4 in the divisor; therefore, we cancel these numbers. Again, 12 is a factor of 36; and 36 ÷ 123, the other factor of 36; therefore, we cancel 12 and 36, and write the quotient above. Lastly, 7 is a factor of 14, and we cancel 7, and write 2, the other factor of 14, below. 11 × 3

16 Ans.

We then have the result

16, Ans. Hence the

2

RULE.-I. Write the factors of the dividend and those of the divisor in form to express the required multiplications and division.

II. Cancel all the factors common to the dividend and divisor. III. Divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor, and the result will be the quotient.

EXAMPLES FOR PRACTICE.

1. What is the quotient of 9 X 18 × 36 × 72 divided by 3 x 6 X 12 X 24?

2. Divide 3 X 4 X 21 X 6 by 8 x 2 x 3 x 7.

3. How many times are 9 times 5 contained in 45 times 15?

4. Divide the product of 13, 26, 48, and 22, by the product of 39, 52, 44, and 11.

5. Divide 3 × 6 × 17 × 8 by 2 × 12 × 51 × 2.

6. Multiply 18 by 3, and divide the product by 3 times 9, multiply this quotient by 36 times 2 divided by 4 times 3.

7. How many bushels of corn, at $1.20 per bushel, must be given for 20 yards of cloth at $1.80 per yard?

8. Find the quotient of

9. Find the quotient of

24 X 15 X 7 × 32 × 27

9 X 16 X 18 X 42.
9 X 16 X 24 X 80
160 X 27 X 48 X 3'
6 x 8 x 4 x 3 x 5
7 X 18 X 12 × 9 × 30°

10. Find the quotient of

11. Divide 16 × 8 × 3 by 19 × 4 × 2 × 2.

12. Divide 11 × 11 × 13 by 22 × 33 × 2.

67. The process of cancellation may be applied to the terms of a proportion.

Take, for example, any proportion, as,

3:6: 12:24.

Any factor common to either ratio, or couplet, may be canceled. Thus, canceling 3 from the first and second terms, we have

1:2:: 12:24.

And, canceling 12 from the third and fourth terms, the result is

1:2::1:2.

Again, equal factors may be canceled from the homologous terms; that is, from the antecedents and consequents. Thus, canceling 3 from the first and third terms, we have

1:6:: 4:24.

Or, canceling 6 from the second and fourth terms, the result is

3:1:12: 4.

NOTE.-For an explanation of the principles involved, and their application, see Proportion.

LEAST COMMON MULTIPLE.

68. A Common Multiple of two or more numbers, is any number which is exactly divisible by each of them. Thus, 32 is a common multiple of 2, 4, 8, and 16.

69. The Least Common Multiple of two or more numbers, is the least number which is exactly divisible by each of them. Thus, 12 is the least common multiple of 2, 3, and 4.

70. Since a product is a multiple of all its factors, it is plain that a common multiple of two or more numbers may be found by multiplying them together; and if the numbers are prime to each other, this product will be their least common multiple.

GENERAL PRINCIPLES.

1. A multiple of a number contains all the prime factors of that number.

2. A common multiple of two or more numbers contains all the prime factors of each of them. Hence,

3. The least common multiple of two or more numbers, is the least number which contains all of the prime factors of each of them.

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

FIRST METHOD.-By Factoring.

EXAMPLE. Find the least common multiple of 21, 18, 12, and 24.

SOLUTION.

21 = 3 ×

18 = 3 × 3 × 2

12 = 3 X 2 X 2

24 = 3 × 2 × 2 × 2

3 X 7 X 3 X2 X2 X2 =

ANALYSIS.-The required multiple must contain all the prime factors of each of the numbers taken separately, and no others. There

504, Ans. fore, it must contain all their different prime factors, each repeated as many times as it is repeated in any of the given numbers. The different prime factors are, 3, 7, and 2. The greatest number of times

that these occur, respectively, in any of the numbers, is 3 twice, 7 once, and 2 three times. Hence, the least common multiple will be composed of two 3's, one 7, and three 2's.

72. Another method of arriving at the same result, is to multiply any one of the numbers (the largest is most convenient) by such prime factors of the other numbers as are not found in that number.

Thus,

24 X 3 X 7504, Ans.

73. SECOND METHOD.-By Division.

EXAMPLE. What is the least common multiple of 8, 16, 18,

ANALYSIS.-Having writ

ten the numbers in a horizontal line, we cancel 8, because it is contained in one of the other numbers. Now, since the remaining numbers are not prime to each other, two or more of them must contain a common prime factor, which must,

of course, be a factor of the required multiple. 2 is such a factor; and dividing by it, we write the quotients underneath. For the same reason, we divide the resulting quotients by 2, and bring down the undivided number 9. In like manner we continue the division, till the quotients and undivided numbers are prime to each other. These, with the divisors, constitute the prime factors of the least common multiple of the given numbers.

THIRD METHOD.-By Cancellation.

74. From the foregoing illustrations and explanations, we deduce the following

PROPOSITION.-The least common multiple of two or more numbers, is the product of any one of the numbers and such prime factors of the other numbers as are not contained in that num

ber.

Hence, the process of cancellation may be applied.

EXAMPLE. Find the least common multiple of 12, 18, 24, 27,

and 36.

SOLUTION.

12 . . 18 . . 24 . . 27 . . 36

ANALYSIS. First, cancel 12 and 18, because

2 X 3 X 36 = 216, Ans. they are factors of 36.

Then, cancel 12, the greatest common factor of 24 and 36, from 24, and write the remaining factor 2, underneath. Lastly, cancel 9, the greatest common factor of 27 and 36, from 27, and write the remaining factor, 3, underneath. The product of these remaining factors and the undivided number, 36, is the least common multiple required.

75. If any of the given numbers can be divided into two or more factors, one of which is contained in a second number, another in a third, and so on, the whole number so divisible must be canceled.

EXAMPLE. What is the least common multiple of 21, 13, 18, 14, and 63?

ber, and is not a factor of any of the other numbers. Now, 18 is composed of two factors, 9 and 2, one of which is cantained in 63, and the other in 14; therefore, cancel 18. Finally, cancel 7, the greatest common factor of 14 and 63, from either number. The remaining factors, 13 × 2 × 63 = 1638, Ans.

NOTE.-Care must be taken, in this process, to avoid canceling a factor too many times. For example, take the numbers 9, 24, and 48. 9 may be canceled, for the reason that one of its factors, 3 and 3, is found in 24, and the other in 48. But, as 3 occurs only once as a factor in either of these numbers, it must not be canceled from either; and the largest other factor which can be canceled is 8, not 12, or 6.

76. From the foregoing examples and illustrations, we derive the following

RULE.-I. Resolve the given numbers into their prime factors, or ascertain their separate prime factors by any of the foregoing methods.