RULE XI.

§ 74. To resolve a COMPOSITE number into its prime factors.

1. Divide the given number by any prime number that will divide it without a remainder; divide the quotient in like manner; and so on, until the quotient becomes a prime number.

2. The several divisors and the last quotient will be the prime factors required.

3. If the given number can only be divided by itself, or a unit, without a remainder, it is itself a prime number.

EXAMPLE.

To resolve 210 into its prime factors.

2)210
3) 105
5) 35

77

The prime divisor 2 resolves 210 into 2X105 (§ 66). The divisor 3 resolves 105 into 3X35; and the divisor 5 resolves 35 into 5×7.

Hence 210 is resolved into the prime factors, 2, 3, 5, and 7.

The application of the preceding Rule will be seen in finding Common Measures, and Common Multiples.

COMMON MEASURE.

$75. One number is called a measure of another, if it is contained in the other an exact number of times, without a remainder.

Thus 2 is a measure of 6, and 3 is also a measure of 6.

Name a measure of 15? Of 35? Of 72? Of 99? Of 132?

If a number can be measured by 2, it is called an even number; otherwise, it is an odd number.

Name all the even numbers to 50. The odd numbers to 55.

$76. A common measure of two or more numbers, is any number which will measure each of them; that is, divide each of them without a remainder.

Thus 3 is a common measure of 12 and 15.

Name a common measure of 18 and 27. Of 20 and 35. Of 32 and 48. Of 12, 18, and 30. Of 36, 54, and 72.

A common measure, it is evident, is any factor which is common to the given numbers; that is, any factor found in each of them.

Greatest Common Measure.

$77. The greatest common measure of two or more numbers, is the greatest number that will measure each of them; that is, divide each of them without a remainder.

Thus 9 is the greatest common measure of 18 and 27.

What is the greatest common measure of 16 and 24? Of 30 and 40? Of 36 and 48? Of 64 and 12? Of 8, 12, and 32?

What is the greatest common measure of 20 and 30? Of 25 and 35? Of 24 and 72? Of 15 and 40? Of 12, 36, and 60?

When two numbers have no common measure greater than a unit, they are said to be prime to each other; thus 16 and 21 are prime to each other.

A common measure is sometimes, though not so properly, called a common divisor; and the greatest common measure, the greatest common divisor.

RULE XII.

§ 78. To find the COMMON MEASURES of two or more numbers.

1. Resolve each number into its prime factors, and select all the factors which are common to the several numbers, that is, which are found in each number.

2. Any one, or the product of any two or more, of these common factors, will be a common measure-and the product of all the common factors will be the greatest common measure of the given numbers.

EXAMPLE.

To find the common measures of 30, 45, and 75.

Resolving each number into its prime factors, we find 30=2×3×5; 45=3×3×5; and 75=3×5×5 (§ 74).

The factors which are common to the three numbers, are 3 and 5; then 3 and 5 are each a common measure, and 3×5,=15, is the greatest common measure of 30, 45, and 75.

It is plain that both 3 and 5 will divide each of the given numbers, without a remainder, as will also 3X5=15; and this last is the greatest number that will so divide them (§ 69).

EXERCISES.

1. Find the greatest common measure of 252,

2. Find the greatest common measure of 120,

3. Find the greatest common measure of 240,

4. Find the greatest common measure of 392,

5. Find the greatest common measure of 504,

6. Find the greatest common measure of 336,

7. Find the greatest common measure of 288,

8. Find the greatest common measure of 460,

9. Find the greatest common measure of 620,

10. Find the several common measures of 42,

180, and 288.

Ans. 36. 144, and 168.

Ans. 24. 336, and 432.

Ans. 48. 504, and 560.

Ans. 56. 567, and 630.

Ans. 63. 588, and 756.

Ans. 84. 480, and 672.

Ans. 96. 1035, and 1150.

Ans. 115. 1116, and 1488.

Ans. 124. 210, and 126,

Ans. 2, 3, 7, 6, 14, 21, and 42.

Another Method of Finding the Greatest Common Measure. § 79. The greatest common measure of the divisor and dividend is the same as that of the divisor and the remainder, if any, after division.

Or, more generally,

§ 80. The greatest common measure of two or more numbers, is the same as that of the least of those numbers and the remainder, or remainders, if any, after dividing the least number into the other, or each of the others.

For example, take the numbers, 12, 28, and 42,

or 12, twice 12+4, and 3 times 12+6.

It is plain that every measure of 12 is also a measure of twice 12, and of 3 times 12; hence no number can measure 12, twice 12+4, and 3 times 12+6, unless it also measures 4 and 6; the greatest common measure, therefore, of 12, 4, and 6, will be the greatest common measure of 12, 28, and 42; and 4 and 6 are the remainders after dividing 12, into 28, and 42. On this principle is founded

RULE XIII.

$ 81. To find the greatest common measure of two or more numbers.

1. For two given numbers,—divide the less number into the greater, and the remainder into the divisor, and the last remainder into the last divisor, and so on, until there is no remainder. The last divisor will be the common measure required.

2. For three or more numbers,—divide the least number into each of the others, and take the remainders and divisor for a new set of numbers, with which proceed as before, and so on, until there is no remainder. The last divisor will be the common measure required.

EXAMPLE.

To find the greatest common measure of 135 and 720.

135)720(5
675

45) 135(3

We divide 135 into 720, and the remainder 45 into the divisor 135, when we get no remainder. The last divisor 45 is the greatest common measure of 135 and 720.

By the principle of § 79, the greatest common measure of 135 and 720 is the same as that of 135 and 45, which is evidently 45. The same principle generalized for two or more numbers, establishes, in like manner, the 2d part of the Rule.

EXERCISES.

1. Find the greatest common measure of 324 and 480.

Ans. 12.

2. Find the greatest common measure of 972 and 1260.

Ans. 36.

3. Find the greatest common measure of 744 and 1680.

Ans. 24.

4. Find the greatest common measure of 480 and 960.

Ans. 480.

5. Find the greatest common measure of 636 and 1080.

Ans. 12.

6. Find the greatest common measure of 375 and 1100.

Ans. 25.

7. Find the greatest common measure of 120 and 1440.

Ans. 120.

8. Find the greatest common measure of 780 and 1560.

Ans. 780.

APPLICATION OF COMMON MEASURE.

1. A farmer has 66 bushels of corn, and 90 of wheat, which he wishes to put into sacks of equal size, and without mixing the two kinds of grain. How many bushels must each sack contain ?

The size of each sack will evidently be any common measure of 66 and 90. Ans. 2 bushels, 3 bushels, or 6 bushels. 2. A gentleman has a corner of ground, the sides of which measure 225 feet, 297 feet, and 369 feet. He wishes to enclose it with a fence having panels of uniform length; what must be the length of each panel? Ans. 9 feet.

3. An upholsterer has 125 yards of carpeting of one kind, 175 of another, and 225 of another. He wishes to divide the whole into pieces of equal length, and the longest that can be obtained; what must be the length of each piece?

Ans. 25 yards.

4. Having 140 acres of land at one place, and 252 at another I wish to divide the whole into fields which shall be of equa size, and the largest that will meet such requisition. What must be the number of acres in each field? Ans. 28 acres.

5. Three regiments of soldiers containing, respectively, 1538 men, 2307 men, and 3845 men, are to be formed, separately, into battalions, the largest that will admit the same number of men in each. What will be the number in each battalion. and the number of battalions in each regiment?

Ans. $769 men in each battalion;

2 battal's in the 1st reg't, 3 in the 2d, and 5 in the 3d.