numbers will be required. The prime numbers in the first fifty are sixteen, viz., 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The following Corollaries enable us readily to discover the Prime Factors of numbers: 1. Every even number is divisible by 2. 2. Every number, whose last two figures express a number which is a multiple of 4, is divisible by 4; for if the number expressed by these two figures is subtracted from the whole number, the remainder will be a certain number of hundreds which are divisible by 4. 3. Every number ending in 5 is divisible by 5. 4. Every number ending with zero is divisible by 10, consequently by 2 and 5. 5. Every number is divisible by 3, when the sum of its figures taken as units is divisible by 3; for if from 1000 one be subtracted, the remainder is divisible by 9; if from 100 one be subtracted, the remainder is divisible by 9; so also if one is subtracted from 10; hence, if from 2000 two be subtracted, the remainder is divisible by 9; 80 also 2 from 200, or 2 from 20; therefore, in dividing by nine, any number of thousands, hundreds or tens, the remainder will always be the unit of the thousands, hundreds, and tens; consequently if the sum of all these remainders as units, and also of the units of the given number, equals 9 or any number of 9's, then the whole number is divisible by 9, and 9 is divisible by 3. PRACTICAL EXAMPLES. 1. A man bought 30 cows at 25 dollars each; he then exchanged the cows for horses at 50 dollars each ; how many horses did he get? Ans. 15 horses. 2. A farmer sold 1500 bushels of wheat at 125 cents per bushel, and received in return barley at 75 cents per bushel. How many bushels of barley did he get? Ans. 2500 bu. barley. 3. How many acres of land, at 25 dollars per acre, can be obtained for 5 houses and lots at 750 dollars each? Ans. 150 acres. 4. How many yards of flannel, three-quarters of a yard wide, will line a coat made of 3 yards of cloth six-quarters wide ? Ans. 6 yards. 5. Sold 320 acres of land at 60 dollars per acre, and invested the proceeds in other land at 40 dollars per acre; how many acres did I get ? Ans. 480 acres. 6. Exchanged 432 pieces of cloth at 18 dollars each, for linen at 6 dollars a piece. How many pieces of linen did I get ? Ans. 1296 pieces. . Sold a farm of 477 acres at 48 dollars per acre, and invested the returns of sale in another farm at $36 per How many acres did I buy ? Ans. 636 acres. 8. Sold 15 horses at 732 dollars each, and bought sheep for the proceeds at 4 dollars each. How many sheep did I buy? Ans. 2745. 9. Multiply the following numbers: 12, 15, 27, 28, 32, and divide the product by 2, 3, 4, 5, 6, 7, 8, and 9. Ans. 12. acre. FRACTIONS. DEF. 1.-If a unit or any other number is divided into equal parts, one or more of these parts is a fraction of the whole, and all the parts constitute the whole. If a unit is divided into two equal parts, each part is called one-half, and is written ; and the two halves constitute the whole; thus, =1. If a unit is divided into three equal parts, each part is one-third (f); two of the parts, f; and the three thirds constitute the whole; thus, = 1. If 5 is divided into two equal parts, each part is five-halves (); two of the parts, 1,0 = 5, etc.; if 6 is divided into three equal parts, each part is f= 2; and two-thirds of 6 is 4, etc. Fourths, fifths, sixths, etc., are similarly constructed. 2. When a unit is divided into equal parts, any number of the parts less than the whole, expressed fractionally, is called a Proper Fraction; as, 1, $, 1, 1%, etc. The quotient is the same in division when the dividend is less than the divisor; as, 1, I, etc.; but when the divisor is less than the dividend, the quotient is called an Improper Fraction; as, , etc. 3. When the division indicated by an Improper Fraction is performed, and the divisor is not contained an exact number of times in the dividend, the quotient is partly integral and partly fractional, and is termed a Mixed Number; thus, = 11; 50 = 8%; and = 104. COR.-The denominator expresses the number of parts into which a unit or any other number is divided, and the numerator expresses the number of parts of a unit taken, or the number divided. EXEMPLIFICATION.-If each half is divided into two equal parts, the whole number of parts is four, and the one-half has made two of those parts; hence, = 4; if each half is divided into three equal parts, the whole number of parts is six, and 1=; .: f==== Po, etc., and j= = = = }, etc.; hence, if both - š t terms of a fraction are multiplied by the same number, the value of the fraction is not changed, and by this principle fractions are reduced to a Common Denominator. PROBLEMS. | 3= 1 X2 (2 = 6 X4=4 1x3=_3 5 x 5=25 X7=28 1. Reduce j and } to a common denominator. , and E. 1 * 12, and . Cor. 1.-The least common multiple of all the denominators is the least common denominator. COR. 2.-Fractions are reduced to a common denominator thus: Multiply both terms of each fraction by the quotient obtained by dividing its denominator into the least common denominator; or, when all the denom GREATEST COMMON DIVISOR. The Greatest Common Divisor of two or more numbers, is the highest number that will exactly divide the numbers. COR.—The greatest common divisor of two or more numbers must be the factor or the product of all the factors which are common to the given numbers. PROBLEMS. Find the greatest common divisor of the following numbers : The only factor common is 2 = G. C. D. 2. Of 12 and 18. 12 = 2, 2, 3, and 18 = 2, 3, 3. One 2 and one 3 are common: therefore, 2 x 3 = 6 = G. C. D. |