6. Paris is in longitude 2° 20' E.; what time is it at Paris when it is noon in London?

7. Boston is in longitude 71° 4′ W.; what time is it in Boston when it is noon at London?

8. New York is in longitude 74° 1' W.; what time is it in New York when it is noon at London?

9. What time is it in Cincinnati, 84° 27′ W., when it is noon in Boston, which is 71° 4′ W.?.

SECTION XII.

DIVISIBILITY OF NUMBERS.

In order to ascertain if a number is divisible by either of the following numbers, 2, 3, 4, 5, 6, 8, 9, 10, or any combination of these, see Sec. VIII, Part I.

To ascertain if a number is divisible by any other number than the above, make trial of other prime divisors, as 7, 11, 13, 17, &c., beginning with the smallest, till you find one that will divide the given number, or find that it is indivisible.

Remember, that in making trial by these numbers, you need not go higher than the square root of the given number, for if a number is divisible, one of the factors will certainly be as small as the square root. Let us take the number 1079; what are its prime factors? By inspection you may see it is not divisible by 2, 3, 5, or 11, consequently not by 4, 6, 8, 9, 10, or 12. On trying it by 7, it is found not divisible by 7; next number is 13; this divides it, giving a quotient, 83, which is prime. Its only factors, therefore, are 13 and 83.

Examples.

1. What are the prime factors of 667?

2. What are the prime factors of 406?

the

3. What are the prime factors of 419? of 361? of 742? of 281? of 316?

4. Prime factors of 941? 812? 749? 1116? 246? 8104? 5. Prime factors of 266? 884? 1917? 376?

6. Reduce

to its lowest terms.

to its lowest terms. In this example it is not evident on inspection whether the two terms of the frac tion have any common divisor. In such cases you may adopt the following Rule to find

The Greatest Common Divisor.

Divide the greater number by the less, and then take the divisor for a new dividend, and divide it by the remainder, and so on, till there is no remainder; the last divisor will be the greatest common divisor.

Apply the above rule to the sixth example.

If the larger number is a multiple of the smaller, it is evident that the smaller is a common divisor of the two numbers; it is also the greatest common divisor; for a number cannot be divided by any number greater than itself; the answer, therefore, is found by the first division. But if there is a remainder, next find whether the remainder will exactly divide the divisor. If it will, it will divide both the original numbers, for if it will divide the divisor, it will divide any multiple of the divisor; and, as it will of course divide itself, it will divide any multiple of the divisor, plus itself. Now the

larger of the original numbers is a certain multiple of the smaller, plus the remainder. If, therefore, after the first division, the remainder will divide the divisor, it is a common divisor, or measure, of the two numbers.

It is also the greatest common divisor; for, as it will exactly measure the smaller of the two numbers, it will exactly measure any multiple of the smaller. Now the greater number, is a certain multiple of the smaller, plus the remainder. The remainder, therefore, in measuring the larger number, is obliged to measure itself. No number greater than itself can do this; therefore the remainder is the greatest common diviIf the work has to be carried on farther than the second division, the same reasoning in the demonstration will apply.

sor.

Examples.

7. What is the greatest common divisor of 874 and 437? 8. What is the greatest common divisor of 497 and 451? 9. What is the greatest common divisor of 817 and 913? 10. What is the greatest common divisor of 1007 and 1219?

11. What is the greatest common divisor of 608 and 192? 12. What is the greatest common divisor of 869 and 1343?

When there are more than two numbers, first find the greatest common divisor of two of them, and then, of that divisor, and the third number.

13. What is the greatest common divisor of 608, 941 and 451?

Whenever it is possible, by inspection, to separate the numbers into their prime factors, this method should be adopted.

14. What is the greatest common divisor of 94, 804 and 126?

15. What is the greatest common divisor of 1274, 896 and 580?

Apply the above Rules to the reduction of the following fractions.

98

16. Reduce to its lowest terms.
17. Reduce to their lowest terms §87, 133, 12?!
18. Reduce to their lowest terms 353, 188, 34.
19. Reduce to their lowest terms fit, ffo, i
tij.

943

To reduce an improper fraction to a whole or mixed number. Perform the division indicated by the fraction as far as possible; if there is a remainder, express that part of the division by placing the denominator under the remainder. 20. Reduce to a whole, or mixed number. Ans. 1. 21. Reduce 27 to a whole, or mixed number. 22. Reduce to a whole, or mixed number, 3, 14, 1†. 23. Reduce to a whole, or mixed number, 9, 73, W. 24. Reduce the improper fractions, 2, 1,

Ans. 33.

SECTION XIV.

CHANGE OF NUMBERS AND FRACTIONS TO HIGHER TERMS.

It is sometimes convenient to express whole numbers in the form of fractions, and to express fractions in higher terms without altering the value. Thus 3, or 4. 10-30, or 0.

Examples.

1. In 4 how many fifths? Ans. 20.

2. Express the value of 4 in fifths. Ans. 2.

3. Express 7 in thirds.

4. Express 19 in the form of sevenths.

5. In 13 how many eighths?

6. Express 21 in thirds.
7. Express 7 in eighteenths.
8. Express 41 in fourths.

9. In 3 how many halves?

10. Change 4 to an improper fraction. 11. Change 17 to an improper fraction.

12. Change 241 to an improper fraction.

13. Change to an improper fraction 183. 1124. 318. 14. Change to eighths, without altering its value.

15. Change to fifteenths.

16. Change to 24ths.

17. Change

to 26ths.

18. Change to 7ths.

This example presents a difficulty, because the required denominator, 7, is not, as in the preceding examples, a multi

ple of the given denominator, 4. We have seen, however, that if we multiply or divide both terms of a fraction by the same number, the value will not be altered. We must then multiply and divide both terms by such numbers as will give us, in the end, 7 for the denominator. The question then is, how can we, by multiplication and division, change 4 into 7? We can multiply it by 7, which will give 28, and then divide by 4, giving 7 for the quotient. Thus the denominator has been changed, by multiplication and division, from 4 to 7. Now whatever has been done to the denominator must be done to the numerator to preserve the value of the fraction. Multiplying 3 by 7, we have 21; dividing this by 4 we have 5 for the required numerator. The answer, therefore, is

51 7

This

fraction, as one of its terms contains a fraction in itself, is called a complex fraction.

19. Change to 8ths. to 9ths. to 7ths.

20. In, how many 4ths? how many 5ths? 6ths?
21. Change 4 to 5ths. 8 to 11ths. 74 to 4ths.
22. Change 22 to 4ths. 184 to 7ths. 31 to 5ths.
23. In 84, how many 3ds? 4ths? 5ths? 9ths?
24. In 19 how many 5ths? 4ths? 7ths?
25. In 94 how many 3ds? 5ths? 8ths?
26. In 134 how many 14ths? 15ths?

27. In 8 how many 17ths? 13ths?

1. A man worked 72 days for § of a dollar a day; what did his wages amount to?

2. Multiply by 46.

3. A man bought 139 bush. of apples for of a dollar a bush.; what did they come to?

4. Multiply by 341. #X127.