PROPOSITION IV. If the denominator of a fraction be divided by any number, the numerator remaining unchanged, the value of the fraction will be increased as many times as there are units in the divisor. Hence, a fraction may be multiplied by a whole number, by dividing the denominator by that number.

Q. If we divide 1 apple into three parts and another into 6, how much greater will the parts of the first be, than those of the second? Are the parts larger as you decrease the denominator? If you divide the denominator by 2, how do you affect the parts? If you divide it by 3? By 4? By 5? By 6? By 7? By 8? Repeat the proposition. How may a fraction be multiplied by a whole number?

§ 84. It appears from Prop. I, that if the numerator of a fraction be multiplied by any number, the value of the fraction will be increased as many times as there are units in the multiplier. It also appears from Prop. III, that if the denominator of a fraction be multiplied by any number, the value of the fraction will be diminished as many times as there are units in the multiplier.

Therefore, when the numerator and denominator of a fraction are both multiplied by the same number, the increase from multiplying the numerator will be just equal to the decrease from multiplying the denominator; hence we have,

PROPOSITION V. If the numerator and denominator of a fraction be multiplied by the same number, the value of the fraction will remain unchanged.

Q. If the numerator of a fraction be multiplied by a number, how many times is the fraction increased? If the denominator be multiplied by the same number, how many times is the fraction diminished? If then the numerator and denominator be both multiplied at the same time, is the value changed? Why not? Repeat the proposition.

EXAMPLES.

49.

1. Multiply the numerator and denominator of by 7 : this gives x=35. Ans. 35. 2. Multiply the numerator and denominator of 12 by 3, by 4, by 5, by 6, by 9, by 12, by 15, by 20.

3. Multiply each term of 125 by 7, by 8, by 12, by 14, by 15, by 17, by 45.

§ 85. It appears from Prop. II, that if the numerator of a fraction be divided by any number, the value of the fraction will be diminished as many times as there are units in the divisor. It also appears from Prop. IV, that if the denominator of a fraction be divided by any number, the value of the fraction will be increased as many times as there are units in the divisor. Therefore, when the numerator and denominator of a fraction are divided by the same number, the decrease from dividing the numerator will be just equal to the increase from dividing the denominator: hence we have,

PROPOSITION VI. If the numerator and denominator of a fraction be divided by the same number, the value of the fraction will remain unchanged.

Q. If the numerator of a fraction be divided by a number, how many times will the value of the fraction be diminished? If the denominator be divided by the same number, how many times will the value of the fraction be increased? If they are both divided by the same number, will the value of the fraction be changed? Why not? Repeat the proposition.

EXAMPLES.

1. Divide both terms of the fraction gives

4) 8

2. Divide each term by 8: this gives 3. Divide each term of the fraction 8, by 16, by 32.

128

by 2, by 4, by

4. Divide each term of the fraction 60 by 2, by 3, by 4, by 5, by 6, by 10, by 12, by 15, by 20, by 30, by 60.

$86. Any number greater than unity that will divide two or more numbers without a remainder is called their common divisor: and the greatest number that will so divide them, is called their GREATEST COMMON DIVISOR.

EXAMPLES.

The greatest

1. Take the two numbers 142 and 994. common divisor cannot be greater than the least number 142. This number will divide itself:-let us see if it will also divide 994.

The number 142 exactly divides itself, giving a quotient of 1; it also divides 994 giving a quotient of 7. Therefore, 142 is the greatest common divisor.

OPERATION.

142)142(1

142

142)994(7 994

The number 2 and 71 are common divisors of the two numbers 142 and 944 since either of them will divide both of the numbers without a remainder. Two numbers may have several common divisors, but they have only one greatest common divisor.

Q. What is the common divisor of two or more numbers? What is their greatest common divisor? What is the difference between the common divisor and the greatest common divisor? What is the common divisor of 2 and 4? Of 4 and 6? What are the common divisors of 4 and 8? What is their greatest common divisor? What are the divisors of 12 and 16? Their greatest common divisor?

2. Take the two numbers 72 and 90.

Let us again see if the least number 72, is the greatest common divisor. After dividing we find a remainder of 18.

OPERATION.

72)90(1
72

greatest common div. 18)72(4 72

Now if 18 will divide 72, it will also divide 90, for 90=72+18, and 18 will be contained once more in 90 72+18 than in 72: but 18 divides 72 without a remainder therefore, 18 is the common divisor: hence we see that the common divisor of two numbers must also be a common divisor between the least number and the remainder after division. But 18 is the greatest common divisor; for, the greatest common divisor must be contained at least once more in 90 than in 72: hence, the greatest

common divisor cannot be greater than the difference between the two numbers, which, in this case is 18. Therefore, we have

PROPOSITION VII. The greatest common divisor of two numbers is obtained by dividing the greater by the less, then dividing the divisor by the remainder, and continuing to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought.

Q. Will the common divisor of two numbers divide their remainder after division? How do you find the greatest common divisor of two numbers?

3. Find the greatest common divisor of the two numbers 63 and 81.

4. Find the greatest common divisor of 315 and 405.

Ans. 45.

5. What is the greatest common divisor of the two numbers 2205 and 2835 ? Ans. 315. 6. Find the greatest common divisor of 1157 and 623 ? Ans. .

7. Find the greatest common divisor of 792 and 1386 ? Ans. 198.

NOTE. If it be required to find the greatest common divisor of more than two numbers, find first the greatest common divisor of two of them, then of that common divisor and one of the remaining numbers, and so on, for all the numbers: the last common divisor will be the greatest common divisor of all the numbers.

8. What is the greatest common divisor of 246, 372 and 522? Ans. 6. 9. What is the greatest common divisor of 492, 744 and 1044?

Ans. 12.

LEAST COMMON MULTIPLE.

§ 87. A number is said to be a common multiple of two or more numbers, when it can be divided by each of them without a remainder. For example, 6 is a common multiple of 2 and 3, because it is exactly divisible by each of them. So likewise, 12 is a common multiple of 2, 3, 4, and 6. because it is divisible by each of them.

The least common multiple of two or more numbers, is the least number which they will separately divide without a remainder. For example, 12 is a common multiple of 2 and 3, but it is not the least common multiple, since 6 is also divisible by 2 and 3. Now 6 being the least number which is so divisible, it is the least common multiple of 2 and 3.

To find the least common multiple of several numbers, we have the following

RULE.

I. Place the numbers on the same line, and divide by the least number that will divide two or more of them without a remainder. and set down in a line below the quotients and the undivided numbers.

II. Divide as before, until there is no number greater than 1 that will exactly divide any two of the numbers: then multiply together the numbers of the lower line, and the divisors, and the product will be the least common multiple. If, in comparing the numbers together we find no common divisor, their product is the least common multiple.

EXAMPLES.

1. Find the least common multiple of 3, 4 and 8.

will not divide 3, we bring down 3 into the 2nd line: we again see that 2 is a common divisor of 2 and 4; and as there is no com

Ans. 2x1×3×2×2=24.

mon divisor between any two of the numbers of the last line, it follows that 2 × 1 × 3 multiplied by the two divisors, is the least common multiple.