8. Divide 1648 by 99.

9. Divide 187432 by 999.

10. Divide 2784567 by 9999.

518. To divide by aliquot parts of 100,

Move the decimal point two places to the left, and multioly the result by 100 divided by the given divisor.

Thus, 84800 ÷ 331 = 848 × 3 = 2544.

519. To divide by aliquot parts of 1000,

Move the decimal point three places to the left, and multiply the result by 1000 divided by the given divisor.

Thus, 647000 ÷ 250 = 647 x 4 = 2588.

DIVISIBILITY OF NUMBERS.

520. Explanation of principles in Art. 94:

(a.) As 2, 4, 6, 8, and 10 are divisible by 2, any number of tens, or any number of tens plus 2, 4, 6, or 8, is divisible by 2; that is, any number whose unit figure is 0 or an even number is divisible by 2.

(c.) As 4 will divide 100, it will divide any number of times 100 plus any number divisible by 4; that is, any number is divisible by 4 when 4 will divide the number expressed by the two right-hand figures.

(d.) As 5 will divide 10, it will divide any number of times 10, or any number of times 10 plus 5; that is, any number whose unit figure is 0 or 5 is divisible by 5.

(g.) As 8 will divide 1000, it will divide any number of times 1000 plus any number divisible by 8; that is, any number is divisible by 8 when 8 will divide the number expressed by the three right-hand figures.

(h.) As 9 is one less than 10, there are as many 9's in any number as there are 10's, and a remainder equal to the number of 10's plus the units figure: thus 10 is 1 nine and 1 remainder; 80 is 8 nines and 8 remainder; 16 is 1 nine and 1 + 6, or 7, remainder; 42 is 4 nines and 4 + 2, or 6, remainder. (See Art. 517.)

7000 contains 777 nines, and a remainder of 7

7 + 6 + 2 + 1, or 16, contains 1 nine and a remainder of 1 + 6, or 7; that is, 7621 contains 845 +1, or 846, nines, and a remainder equal to the excess over 9 of the sum of its figures. In like manner, if any number is divided by 9, the remainder is equal to the excess over 9 of the sum of its figures. Hence, any number is divisible by 9 when the sum of its figures is divisible by 9.

NOTE 1. This explains the method of proof in addition, subtraction, etc., called "casting out the 9's."

(b.) As 9 is divisible by 3, any number of times 9 plus 3, or 6, will be divisible by 3; that is, any number is divisible by 3 when the sum of its figures is divisible by 3.

(e.) As 3 is not an even number, and every even number is divisible by 2, it is evident that any even number divisible by 3 is divisible by 6.

NOTE 2. Every prime number, but 2 and 5, has 1, 3, 7, or 9 for its unit figure.

521. For further aid in determining the factors of numbers, we have the following

TABLE OF PRIME NUMBERS FROM 1 TO 997.

GREATEST COMMON DIVISOR.

522. To explain the method of finding the greatest common divisor of numbers given in Art. 100, the following principles are given :

1. A common divisor of two numbers is also a common divisor of the sum or the difference of any multiples of each.

Thus, 16 is divisible by 4; it is evident that any other number less, or greater, than 16 by any integral number of 4's, is also divisible by 4. It therefore follows that 4 will divide any multiple of 16 plus or minus any number of 4's.

The same method of reasoning applies to all numbers.

2. The greatest common divisor of two numbers is also the greatest common divisor of the less and the remainder after dividing the greater by the less.

Suppose it is required to find the greatest common divisor of 27 and 21.

21) 27 (1
21

6

According to the first principle stated above, as 27=21 + 6, any divisor of 21 and 6 must be a divisor of 27; and as 6=27 - 21, any divisor of 27 and 21 must be a divisor of 6; that is, the divisors of 27 and 21 and of 21 and 6 are identical, and therefore the greatest common divisor of 27 and 21 must also be the greatest common divisor of 21 and 6.

In like manner, the greatest common divisor of 21 and 6 is the greatest common divisor of 6 and the remainder after dividing 21 by 6.

The same method of reasoning applies to all numbers. Hence we derive the rule given in Art. 100, page 65.

523. To add or subtract two fractions whose numerators are each unity,

Rule.

To add take the sum of the denominators, to subtract take the difference of the denominators, for a numerator, and in either case take the product of the denominators for a denominator.

524. GREATEST COMMON DIVISOR OF FRACTIONS. 15. Find the greatest common divisor of 3, §, and .

Solution. The greatest common divisor of the numerators, 2, 4, and 8, is 2; and the greatest common divisor of thirds, ninths, and fifteenths is forty-fifths; that is, is a fraction whose numerator is 1, and denominator the least common multiple of the denominators. Therefore the greatest common divisor of 3, 4, and is. Hence,

525. To find the greatest common divisor of two or more fractions,

Rule.

Divide the greatest common divisor of the numerators by the least common multiple of the denominators.

Find the greatest common divisor of

(16.),, and .

(17.) 1, 2, and §.

(18.) 74, 23, and 8.

(19.) 13, §, and .

(20.) 21, 51, 61, and 43.

526. LEAST COMMON MULTIPLE OF FRACTIONS.

21. Find the least common multinle of,, and 21.

Solution. The least common multiple of the numerators, 2, 4, and 8, is 8; and the least common multiple of ninths, fifteenths, and twenty-firsts, is thirds; that is, is a fraction whose numerator is 1 and denominator the greatest common divisor of the denominators. Therefore the least common multiple of 2,, and ♬ is §, or 2}. Hence,

527. To find the least common multiple of two or more fractions,

Rule.

Divide the least common multiple of the numerators by the greatest common divisor of the denominators.

Find the least common multiple of (22.), o, and 1.

(23.) 2, 4, and 63.

(24.) 11, 7, 1.

(25.) 2, 3, 4, and 6.

(26.) 2, 5, 8, and 91.

27. If A and B start from the same point, at the same time, to walk round an island, and A can walk round it in of a day, and B in 3 of a day, how many days will it be before they will be together again? How many times round the island must each go?

CIRCULATING DECIMALS.

528. Circulating Decimals are decimals in which the same figure, or set of figures, is repeated without limit. Thus, if we attempt to reduce to a decimal, we have 0.33333, etc., the 3's repeating without limit. 40.363636, etc., the 36 repeating without limit Instead of repeating the fig