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In finding factors common to two or more numbers the following principles are of service.

1. A factor of a number is a factor of any multiple of that number.

2. A common factor of two numbers is a factor of the sum or difference of any of their multiples.

The greatest common divisor of even very large numbers can sometimes be readily found by the application of the principles just stated.

Fractions Reduced to Lowest Terms

A fraction is said to be in its lowest terms when its numerator and denominator contain no common factor other than 1. Thus, į and are in their lowest terms, but it is not, as its numerator and denominator contain a common factor 6.

Rule. — To reduce a fraction to its lowest terms, divide numerator and denominator by their common factors.

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6

8

Reduce to lowest terms the following fractions : 1. 22

6.

11. 11 2.

7.

12.14 3. 8. 4

13. 34 4. 11

9. 13

14. l 5. 10.13

15.

8 24

9

36 108

22

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11.

Exercise Reduce to lowest terms the following fractions : 1. 1992

21. 140 2. 1932

22. 3. 19%

13. 25 4. 194 14. 144

24. 24 5. 18

96

288 12. 108

288

196 84 196 64 2 56

23.

15. 160

240

84 140

2 88

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The Least Common Multiple of two or more numbers is the smallest number which will contain them all without a remainder. It is usually abbreviated L. C. M.

24 is the least common multiple of 3, 8, and 12, because 24 is the smallest number which is divisible by all of the given numbers without a remainder.

Considerable use is made of least common multiple in the addition and subtraction of fractions, but the work is generally simple. Much of the labor in finding the least common multiple can be avoided by determining

whether some of the numbers given are factors of any of the others,

Rule. — (1) Separate each number into its prime factors.

(2) Find the product of these factors, using each factor the greatest number of times it occurs in any one of the given numbers.

EXAMPLE. Find the least common multiple of 2, 7, 3, 14, 15, and 30. As 30 is divisible by 15, the 15 need not be any longer considered, for any number that is divisible by 30 is also divisible by 15. In like manner the 2 and 3 can be disposed of, for they also are contained in 30. Similarly 7 is a divisor of 14 and hence 7 need not be further considered. The problem has now resolved itself into finding the least common multiple of 14 and 30. Either of the following arrangements can be used. 14 = 2 x 7

2/14 30 30 2 X 3 X 5

7 15 L. C. M. = 2 X 7 X 3 X 5= 210 L. C. M.=2 x 7 x 15 = 210

ORAL DRILL Find the Least Common Multiple of the numbers given in each of the following examples : 1. 2, 3, 5 11. 3, 9, 15

21. 44, 8, 88
2. 3, 4, 5
12. 3, 9, 45

22. 11, 121, 33
3. 3, 5, 9
13. 3, 9, 54

23. 4, 60, 20
4. 4, 6, 12
14. 3, 9, 63

24. 8, 72, 3
5. 4, 6, 16
15. 4, 5, 6

25. 8, 56, 16
6. 4, 6, 20
16. 12, 16

26. 11, 22, 33
7. 6, 8, 12
17. 40, 50

27. 24, 36, 72
8. 6, 8, 16
18. 50, 210

28. 24, 72, 96
9. 6, 8, 20
19. 75, 300

29. 24, 72, 240 10.6, 8, 24 20. 21, 35

30. 12, 144, 1728

1

Exercise Find the Least Common Multiple of the numbers given in each of the following examples :

1. 16, 24, 40 11. 18, 54, 90 21. 88, 16, 176 2. 24, 32, 40 12. 18, 54, 270 22. 22, 242, 66 3. 24, 40, 72 13. 18, 54, 324 23. 8, 120, 40 4. 32, 48, 96 14. 18, 54, 378 24. 16, 144, 6 5. 32, 48, 128 15. 24, 30, 36 25. 16, 112, 32 6. 32, 48, 160 16. 72, 96

26. 22, 44, 66 7. 48, 64, 96 17. 240, 300 27. 48, 72, 144 8. 48, 64, 128 18. 300, 1260

28. 48, 144, 192 9. 48, 64, 160 19. 450, 1800 29. 48, 144, 480 10. 48, 64, 192 20. 126, 210

30. 24, 288, 3456

Addition of Fractions

The fundamental principle in adding fractions is that the demoninators must be equal.

Common Denominator. — If two or more fractions have the same denominator they are said to have a common denominator. Thus, ia and 12 have the common denominator 12.

Similar Fractions. - Fractions that have a common denominator are called similar fractions.

Rule. - To add fractions, (1) reduce them to similar fractions, (2) write the sum of the numerators over the least common denominator, and (3) reduce the result to lowest terms.

To add mixed numbers, add separately the whole numbers and the fractions and find the sum of the results.

EXAMPLE. Add 23, 32, 45.

23 = 24
3} = 38

98 = 107 or 101, the sum.

Exercise In each of the following examples find the sum of the numbers given. 1. , 7, 11

9. }, }, }, }, }, + 2. , , , 11, 14, 19 10. 18, 26, 43 3. ,

11. 31, 71, 83 4. , , ,

12. 163, 87, 93, 37, 13 5. ^, , 12,5

13. 110, 27, 5} 6. 11, 22, 13,

14. 216, 33, 41 7. V, 34, in,

15. 316, 33, 34 8. 214, 17, 18, 31%

16. 51, 43, 3112

4 9

13 13 19 233 234 233 234 235 23

107 4, 144, 144, 144

Prime Denominators. When the denominators of two or more fractions are prime to each other observe the following rule.

Rule. — (1) Multiply the denominator of each fraction by the numerator of the other, (2) add these products, and (3) write it over the product of the denominators. EXAMPLE. Add and .

2 x 5= 10

3 x 4 12
Their sum= 22, the numerator of the answer ;

5 X 3 = 15, the denominator of the answer.
Then plus is 13, or 17.

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