Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

104. A number that is a factor of two or more numbers is called a Common Divisor of them.

Thus, 5 is a common divisor of 10 and 15.

105. The greatest factor of each of two or more numbers is called the Greatest Common Divisor of them.

Thus, 6 is the greatest common divisor of 18 and 24.

106. When two or more numbers have no common factor or divisor, they are Prime to each other.

Thus, 8 and 15 are prime to each other.

107. PRINCIPLE.

[ocr errors]

The greatest common divisor of two

or more numbers is the product of all their common prime factors.

Written.

1. What is the greatest common divisor of 90 and 150.

[blocks in formation]

108. To find the greatest common divisor when the numbers cannot be readily factored.

17. What is the greatest common divisor of 510 and 935?

[ocr errors]

SOLUTION. The greatest common divisor must be a factor of both these numbers. It cannot be the larger.

510)935(1
510

425)510(1
425

It is not the smaller, for we find a remainder of 425 after dividing the larger by the smaller.

If the remainder 425 is a factor of 510, it will be the greatest common divisor of 425 and 510, and therefore of

Greatest Common Divisor 85)425(5 510 and 935. But it is not,

425

for we find a remainder of 85 after dividing 510 by 425.

If the remainder 85 is a factor of 425, it will be the greatest common divisor of itself and 425, also of 425 and 510; also of 510 and 935. We find that 85 is a divisor of 425. It is therefore the greatest common divisor of 510 and 935.

NOTE 1. An exact divisor of a number is an exact divisor of any number of times that number.

NOTE 2. —An exact divisor of each of two numbers is an exact divisor of their sum and of their difference.

Rule. · Divide the greater number by the smaller, and the last divisor by the last remainder until there is no remainder. The last divisor will be the greatest common divisor. If more than two numbers are given, find the greatest common divisor of two of them, then of this divisor and a third number, and so on. The last divisor will be the greatest common divisor.

Find the greatest common divisor:

[blocks in formation]

30. Find the greatest common divisor of 72, 153, 315,

29. 225, 360, 405

2187.

31. I have 32 bushels of wheat, 48 of barley, and 128 of oats. I desire to put all this grain into boxes of the largest possible size, so that no box shall contain more than one kind of grain. How many bushels must each box contain?

There will be how many boxes of wheat? Of barley? Of oats?

LEAST COMMON MULTIPLE.

109. 1. Name a number of which 3 is a factor. Of which 5 is a factor.

2. Name several numbers that are exactly divisible by 2. By 7. By 5.

3. Name a number that is exactly divisible by both 3 and 2. Name another. Another. Another. What is the least number that is exactly divisible by 3 and 2?

4. What is the smallest number that will exactly contain 5 and 6?

110. A Multiple of a number is a number that exactly contains it.

Thus, 5, 10, and 15 are multiples of 5.

NOTE.

Pupils sometimes mistake multiples for factors.

A multiple is a product. A factor is a divisor.

111. A Common Multiple of two or more numbers is any number that exactly contains each of them.

Thus, 60 is a common multiple of 4, 5, and 6.

112. The Least Common Multiple of two or more numbers is the smallest number that exactly contains each of them. Thus, 30 is the least common multiple of 3, 5, and 6.

113. PRINCIPLE.

The least common multiple of two or more numbers is the product of all the prime factors in the largest number multiplied by the product of such prime factors of the other numbers as are not found in the largest.

114. Written.

1. What is the least common multiple of 21, 28, and 30? SOLUTION. Separating the numbers into their prime factors, and multiplying the product of the prime factors of the largest number,

[ocr errors]

21= 3 x 7
28= 2 × 2 × 7
30= 2 x 3 x 5

2 × 3 × 5 × 2 × 7 = 420.

2 × 3 × 5 = 30, by the product of the prime factors of the other numbers not found in the largest, we have 2 × 7=14. Therefore 14 x 30 420, the least common multiple. The prime factors that enter into this least common multiple are 2, 3, 5, 2, 7..

The least common multiple must contain all the factors of 30 (2 × 3 × 5) or it would not contain 30. It must contain the prime factors of 21 (3 × 7). 3 is also a prime factor of 30, and is not again included, but 7, not being a factor of 30, must be included in the least common multiple, or it will not contain 21.

Of the prime factors of 28 (2 × 2 × 7), there are two 2's. Since the largest number has but one factor 2, the factor 2 must be again included in the least common multiple, or it would not contain 28. The factor 7 is excluded because it is also a factor of 21. We now find that all the factors of the three numbers are found among the factors of the least common multiple 2 × 3 × 5 × 2 × 7.

The practical method of finding the least common multiple is as follows:

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[blocks in formation]

9. 15, 60, 140, 210

13. 24, 48, 96, 192

5. 21, 27, 36 14. Find the contents of the smallest box that may be filled with wheat by using a 4-quart, a 5-quart, or a 6-quart measure. How many 4-quart measures will fill it? How many 5-quart measures? 6-quart?

15. Three boys ride around a circular track. A goes around once in 5 minutes, B once in 8 minutes, C once in 10 minutes. If they start together, how many minutes must elapse before they all come together at the startingpoint? How many times will each have gone around the circle?

115. Review of Factors, Multiples, Divisors, and CancelZation.

1. Define factor, composite number, prime number, and prime factor.

2. Find the prime factors of 5075; of 9576; of 3150; of 6006.

3. Find the sum of the prime factors of 34650.

4. Find the prime factors of 2310; of 17199; of 6840.

5. 81158 is the product of what prime factors?

6. Find the largest prime factor of 12600.

7. What is a common divisor of two or more numbers?

8. What is the greatest common divisor of two or more numbers?

9. When are numbers prime to each other?

« ΠροηγούμενηΣυνέχεια »