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number of times. For a like reason 60 is a multiple of 12, since it contains 12 an exact number of times.

FIRST PRINCIPLE. Every number which exactly di. vides another number will also divide without a remainder any multiple of that number. For example, 24 is divisible by 8, giving a quotient 3. Now, if 24 be mul. tiplied by 4, 5, 6, or any other number, the product so arising will also be divisible by 8.

SECOND PRINCIPLE. If a number be separated into two parts, any divisor which will divide each of the parts separately, without a remainder, will exactly divide the given number. For, the sum of the two partial quotients must be equal to the entire quotient; and if they are both whole numbers, the entire quotient must be a whole number; for the sum of two whole numbers cannot be equal to a fraction.

For example, if 36 be separated into the parts 16 and 20, the number 4, which will divide both numbers 16 and 20, will also divide 36 ; and the sum of the quotients 4 and 5 will be equal to the entire quotient 9.

THIRD PRINCIPLE. If a number be decomposed into two parts, then any divisor which will divide the given number and one of the parts, will also divide the other.

For, the entire quotient is equal to the sum of the two partial quotients; and if the entire quotient and one of the partial quotients be whole numbers, the other must also be a whole number; for no proper fraction added to a whole number can give a whole number.

Let it be required to find the greatest common divisor of the numbers 216 and 408.

It is evident that the greatest common divisor cannot be

216)408(1 greater than the least number

216 216. Now, as 216 will divide

192)216(1 itself, let us see if it will divide

192 408; for if it will, it is the

24)192(8 greatest common divisor sought.

192 Making this division, we find a quotient 1 and a remainder 192; hence, 216 is not the

OPERATION.

greatest common divisor. Now we say, that the greatest common divisor of the two given numbers is the common divisor of the less number 216 and the remainder 192 after the division. For, by the second principle, any number which will exactly divide 216 and 192, will also exactly divide the number 408.

Let us now seek the common divisor between 216 and 192. Dividing the greater by the less, we have a remainder of 24; and from what has been said above, the greatest common divisor of 192 and 216 is the same as the greatest common divisor of 192 and 24, which we find to be 24. Therefore, 24 is the greatest common divisor of the given numbers 408 and 216: hence, to find the greatest common divisor,

Divide the greater number by the less, and then divide the divisor by the remainder, and continue to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought.

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.

EXAMPLES.

1. Find the greatest common divisor of 408 and 740.

Ans. 4. 2. Find the greatest common divisor of 315 and 810.

3. Find the greatest common divisor of 4410 and 5670.

Ans. 630. 4. Find the greatest common divisor of 3471 and 1869.

Ans. 267. 5. Find the greatest common divisor of 1584 and 2772.

Ans. 6. What is the greatest common divisor of 492, 744, and 1044 ?

Ans. 12.

7. What is the greatest common divisor of 944, 1488, and 2088 ?

Ans. 8. 8. What is the greatest common divisor of 216, 408, and 740 ?

Ans. 4. 9. What is the greatest common divisor of 945, 1560, and 22683 ?

Ans. 3. 10. What is the greatest common divisor of 204, 1190, 1445, and 2006 ?

Ans.

LEAST COMMON MULTIPLE.

125. A number is said to be a common multiple of two or more numbers, when it can be divided by each of them separately, 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 mul. tiple of 2 and 3.

A factor of a number is any number that will divide it without a remainder; and a prime factor is any prime number which will so divide it.

126. To find the least common multiple of several numbers,

I. Place tke numbers on the same line, and divide by any prime 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.

125. When is one number said to be a common multiple of two or more numbers? Of what numbers is 6 a common multiple ? Of what numbers is 8 a common multiple? What is the least common multiple of two or more numbers? What is the difference between a common multiple and the least common multiple? What is a factor of any number? What is a prime factor?

II. Then 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 mul. tiple.

EXAMPLES.

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1. Find the least common multiple of 3, 4, and 8. We first see, that 2 will

OPERATION. divide 4 and 8, but as it

2)3 ...4

.8 will not divide , we bring

2)3 ...2

4 down 3 into the 2d line :

3 . 1 2 we again see that 2 is a common divisor of 2 and Ans. 2X1 X3 X2 X 2=24.

and as there is no com. mon divisor between any two of the numbers of the last line, it follows that 2x1 x3 multiplied by the two divisors, is the least common multiple.

2. Find the least common multiple of 3, 8, and 9, We arrange the numbers in a

OPERATION. line, and see that 3 will divide 3)3 ...8...9 two of them. We then write down

1 8...3 the quotients 1 and 3, and also the

1x8X3X3=72. 8, which cannot be divided. Then as there is no common divisor between any two of the numbers, 1, 8, and 3, it follows that their product, multiplied by the divisor 3, will give the least common multiple sought. 3. Find the least common multiple of 6, 7, 8, and 10.

Ans. 840. 4. Find the least common multiple of 21 and 49.

Anrs. 147. 5. Find the least common multiple of 2, 7, 5, 6, and 8.

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Ans. 840.

126. Give the rule for finding the least common multiple. If the numbers have no common divisor, what is the least common multiple ?

6. Find the least common multiple of 4, 14, 28,

and 98.

Ans. 7. Find the least common multiple of 13 and 6.

Ans. 78. 8. Find the least common multiple of 12, 4, and 7.

Ans. 84. 9. Find the least common multiple of 6, 9, 4, 14, and

Ans. 1008. 10. Find the least common multiple of 13, 12, and 4.

Ans. 156. 11. What is the least common multiple of 11, 17, 19, 21, and 7?

Ans. 74613.

16.

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REDUCTION OF VULGAR FRACTIONS.

127. Reduction of Vulgar Fractions is the method of changing their forms without altering their value.

A fraction is said to be in its lowest terms when there is no number greater than 1 that will divide the numerator and denominator without a remainder. The terms of the fraction are then said to have no common factor.

CASE I.

128. To reduce an improper fraction to its equivalent whole or mixed number,

Divide the numerator by the, denominator, the quotient will be the whole number ; and the remainder, if there be one, placed over the given denominator, will form the frac

tional part.

127. What is reduction ? When is a fraction said to be in its lowest torms? Is one-half in its lowest terms? Is two-fourths? Is three-fourths ?

128. How do you reduce a fraction to its equivalent whole or mixed number? Does this reduction alter its value? Why not? What is four-halves equal to? Eight-fourths ? Sixteen-eighths ? Twenty fifths ? Twenty-six sixths ? Four-thirds? What is nine. fourths equal to? Five-fourths ? Seventeen-sixths ? Eighteen-sovonths ?

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