DEM.-Take any two numbers, as 12 and 20, of which 4 is a common divisor. Now, 12 equals three times 4, and 20 equals five times 4, and their sum equals three times 4 plus five times 4, or eight times 4, which contains 4, or is divisible by 4. Their difference is five times 4 minus three times 4, or two times 4, which is also divisible by 4.

CASE I.

120. When the numbers are small and can be readlly factored.

FIRST METHOD.

121. This method consis.s in finding the common factors, and taking their product.

1. Find the greatest common divisor of 42,

84,, and 126.

OPERATION. 214284 - 126

321

7 7

42

63

14 21

SOLUTION.-We write the numbers one beside another, as in the margin. Dividing by 2, we see that 2 is a factor of each number; it is therefore a factor of the G. C. D. (Prin. 1). Dividing the quotients by 3, we see that 3 is a factor of each number and therefore a factor of the G. C. D.; and in the same way we see that 7 is a factor of the G. C. D. Now since the quotients 1, 2, and 3 are prime to each other, 2, 3, and 7 are all the common factors; hence their product, which is 42, is the G. C. D. (Prin. 2).

1 2

3

2×3×7=42

Rule.-I. Write the numbers one beside another, with a vertical line at the left, and divide by any common factor of all the numbers.

II. Divide the quotients in the same manner, and thus continue until the quotients have no common factor.

II. Take the product of all the divisors; the result will

be the greatest common divisor.

What is the greatest common divisor of

SECOND METHOD.

122. This method consists in resolving the numbers into their prime factors, and taking the product of the common factors.

1. Find the greatest common divisor of 42, 84, and 126.

SOLUTION.-The factors of 42 are 2, 3, and 7; the factors of 84 are 2, 2, 3, and 7; the factors of 126 are 2, 3, 3, and 7. We see that 2, 3, and 7 are all the prime factors common t the three numbers; hence their product, which is 42, is the greatest common divisor of the numbers (Prin. 2). Hence the following

OPERATION.

42=2×3×7 84=2x2x3x7 126=2×3×3x7 2×3×7=42

Rule.-Resolve the numbers into their prime factors, and take the product of all the common factors.

Find the greatest common divisor of

2. 270, 315, and 405.

3. 168, 192, and 216.

4. 252, 308, and 364.

5. 504, 546, and 588. 6. 392, 448, and 504.

7. 432, 504, and 648.

8. 792, 864, 936, and 1008.

Ans. 45.

Ans. 24.

Ans 28.

Ans 42.

Ans. 56.

Ans. 72.

Ans. 72.

Ans. 192.

9. 384, 576, 768, and 960.

CASE II.

123. When the numbers are large and cannot be readily factored.

1. Find the greatest common divisor of 32 and 56.

SOLUTION. We divide 56 by 32, the divisor 32 by 24, and the divisor 24 by the remainder 8, and have no remainder; then will 8 be the greatest common divisor of 32 and 56. For

1st. The last remainder, 8, is a NUMBER OF TIMES the G. C. D. Since 32 and 56 are each a number of times the G. C. D., their difference, 24, is a number of times the G. C. D., by Prin. 3; and since 24 and 32 are each a number of times the G. C. D., their difference, 8, is also a number of times the G. C. D.

OPERATION.

32)56(1
32
24)32(1
24

8)24(3 24

2d. The last divisor, 8, is ONCE the G. C. D. Since 8 divides 24 it will divide 24+8, or 32, by Prin. 3; and since it divides 32 and 24, it will divide 32+24, or 56; and now since 8 divides 32 and 56, and is a number of times the G. C. D., it must be once the G. C. D.

ANOTHER FORM.-In the margin on the right is another form of writing the division, which in practice we prefer to the above. The problem is to find the greatest common divisor of 32 and 116. method will be clear from a slight inspection of the work. Let the pupils adopt it after they are familiar with the common form.

The

Rule.-Divide the greater number by the less, the divisor by the remainder, and thus continue to divide the last divisor by the last remainder until there is no remainder; the last divisor will be the greatest common divisor.

NOTE. To find the greatest common divisor of more than two numbers, we first find the greatest common divisor of two of them, then of that divisor and one of the other numbers, etc.

6. A farmer has two heaps of apples, one containing 364, and the other 585, which he wishes to divide into smaller heaps, each containing the same number; what is the largest number that the heaps may contain?

Ans. 13.

7. A benevolent society distributed $678, $906, and $1146 in equal sums to the poor of three wards of a city, the sums being as large as possible. Required the amount of the equal sums and the number of persons receiving relief in each ward?

12. A Western landholder has three tracts, the first containing 583 acres, the second 574 acres, and the third 861 acres, which he wishes to divide into fields of equal size, having the least number possible. Required the number of fields and the number of acres in each. Ans. 48 fields; 41 acres.

INTRODUCTION TO COMMON MULTIPLE.

MENTAL EXERCISES.

1. What number is three times 5? four times 6? five times 6? six times 8?

2. A number which is one or more times another number is called a multiple of that number.

3. What is the multiple of 4? of 5? of 6? of 7 of 8 of 9? c* 10? of 11 of 12?

4. What multiple is common to 2 and 3? to 3 and 4? to 4 and 6? to 6 and 8 to 6 and 9?

5. What may we call a multiple common to two or more numbers! Ans. A common multiple.

6. What is a common multiple of 4 and 5? 8 and 9? 6 and 7? 4 and 65 and 8?

7. What is the least multiple common to 2 and 4? 4 and 6? 4 and 8? 6 and 8? 8 and 12?

8. What shall we call the least multiple common to two or more numbers? Ans. Their least common multiple.

9. What is the least common multiple of 4 and 6? 9 and 12? 10 and 16 20 and 24? 25 and 30?

LEAST COMMON MULTIPLE.

124. A Multiple of a number is one or more times the number; thus, 4 times 5, or 20, is a multiple of 5.

125. A Common Multiple of two or more numbers is a number which is a multiple of each of them; thus, 24 is a common multiple of 2, 3, and 4.

126. The Least Common Multiple of two or more numbers is the least number which is a multiple of each of them; thus, 12 is the least common multiple of 2, 3, and 4.

NOTE.-The least common multiple may be represented by the initials L. C. M.

PRINCIPLES.

1. A multiple of a number is exactly divisible by that number.

2. A multiple of a number must contain all the prime factors of that number.

3. A common multiple of two or more numbers must con tain all the prime factors of each of those numbers.

4. The least common multiple of two or more number: must contain all the prime factors of each number, and no other factors.

CASE I.

127. When the numbers are small and easily fac tored.

FIRST METHOD.

128. This method consists in resolving the numbers into their prime factors, and taking the product of all the differ. ent factors."

1. Find the least common multiple of 12, 30, and 70.

OPERATION.

12=2×2×3
30=2x3x5

70=2X5×7

2×2×3×5x7=420

SOLUTION. We first resolve the numbers into their prime factors. A multiple of 12 must contain the factors of 12, 2, 2, 3; a multiple of 30 must contain the factors of 30, 2, 3, 5; a multiple of 70 must contain the factors of 70, 2, 5, 7; hence the common multiple of 12, 30, and 70 must contain all these different factors and no others; therefore 2×2×5×3×7, or 420, is the L. C. M. of 12, 30, and 70 (Prin. 4).

Rule.-I. Resolve the numbers into their prime factors. II. Take the product of all the different factors, using each factor the greatest number of times it occurs in either number.

NOTE. Any numbers which are divisors of the others may be omitted, since the multiple of the other numbers will be a multiple of these.

Find the least common multiple of

2. 24, 30, and 36.

3. 16, 24, and 56.

4. 28, 36, and 60.
5. 36, 48, and 84.
6. 63, 72, and 108.
7. 15, 30, 42, and 72.

8. 22, 55, 77, and 110.

Ans. 360.

Ans. 336.

Ans. 1260.

Ans. 1008.

Ans. 1512.

Ans. 2520.

Ans. 770.

9. 33, 99, 36, 108, and 135.

Ans. 5940.

10. A has $14, B $15, C $36, and D as many as the least

common multiple of the amounts of the others; how many bas D?

Ans. $1260.