13. The product of two numbers is 144, and one of the numbers is 8: what is the other number? Solution.—144-8=18, the required number. (Art. 120.) 14. The product of A and B's ages is 3250 years, and B's age is 50 years : what is the age of A ? 15. The product of the length of a field multiplied by its breadth is 15925 rods, and its breadth is 91 rods : what is its length ? 157. When the divisor and quotient are given, to find the dividend. Multiply the given divisor and quotient together, and the product will be the dividend. (Art. 121.) 16. If a certain divisor is 12, and the quotient is 30, what is the dividend ? Solution.-30X12=360, the dividend required. 17. If the quotient is 275 and the divisor 683, what must be the dividend ? 18. If the divisor is 1031 and the quotient 1002, what must be the dividend ? 158. When the dividend and quotient are given, to find the divisor. Divide the given dividend by the given quotient, and the quotient thus obtained will be the number required. (Art. 122.) 19. A certain dividend is 864, and the quotient is 12 : what is the divisor ? Solution.--864-12=72, the divisor required. (Art. 120.) 20. A gentleman handed a purse containing 1152 shillings, to Quest.-157. When the divisor and quotient are given, how is the dividend found ? 158. When the dividend and quotient are given, how is the divisor found ? a company of beggars, which was sufficient to give them 24 shillings apiece: how many beggars were there? 21. A farmer having 2500 sheep, divided them into flocks of 125 each : how many flocks did they make ? 159. When the product of three numbers and two of the numbers are given, to find the other number. Divide the given product by the product of the two given numbers, and the quotient will be the other number. 22. There are three numbers whose product is 288; one of them is 8, and another 9 : it is required to find the other number. Solution.-9 X 8=72; and 288: 72=4, the number required. PROOF.—9 X 8X4=288, the given product. 23. The product of three persons' ages is 14880 years ; the age of the oldest is 31 years, and that of the second is 24 years : what is the age of the youngest ? 24. If a garrison of 75 men have 18750 pounds of meat, how long will it last them, allowing 25 pounds to each man per month? 25. The sum of two numbers is 3471, and the less is 1629 : what is the greater ? 26. The sum of two numbers is 4136, and the greater is 3074: what is the less ? 27. The difference between two numbers is 128, and the greater is 760: what is the less ? 28. The difference between two numbers is 340, and the less is 634: what is the greater ? 29. The sum of two numbers is 12640, and their difference is 1608: what are the numbers ? 30. The sum of two numbers is 25264, and their difference is 736 : what are the numbers ? 31. The sum of two numbers is 42126, and their difference is 176 : what are the numbers ? 32. The product of two numbers is 246018, and one of them is 313: what is the other number? SECTION VI. PROPERTIES OF NUMBERS.* Art. 160. The progress as well as the pleasure of the student in Arithmetic, depends very much upon the accuracy of his knowledge of the terms, which are employed in mathematical reasoning. Particular pains should therefore be taken to understand their true import. DEF. 1. An integer signifies a whole number. (Art. 28. Obs. 2.) 2. Whole numbers or integers are divided into prime and composite numbers. 3. A composite number, we have seen, is one which may be produced by multiplying two or more numbers together; as, 4, 10, 15. (Art. 95.) 4. A prime number is one which cannot be produced by multiplying any two or more numbers together ; or which cannot be exactly divided by any whole number, except a unit and itself. Thus, 1, 2, 3, 5, 7, 11, 13, &c., are prime numbers. Obs. 1. One number is said to be prime to another, when a unit is the only number by which both can be divided without a remainder. 2. The learner must be careful not to confound numbers which are prime to each other with prime numbers; for numbers that are prime to each other, may themselves be composite numbers. Thus 4 and 9 are prime to each other, while they are composite numbers. 3. The number of prime numbers is unlimited. For those under 3413, see Table, page 94. 5. An even number is one which can be divided by 2 without a remainder ; as, 4, 6, 8, 10. Quest.-160. Upon what does the progress and pleasure of the student in Arithmetic very much depend? What is an integer ? What is a composite number? What is a prime number? Are prime numbers divisible by other numbers ? Obs. When is one arimber said to be prime to another ? How many prime numbers are there? What is an even pumber? An odd number? Obs. Are even numbers prime or composite? What is true of odd numbers in this respect? * Barlow on the Theory of Numbers ; also, Bonnycastle's Arithmetio. 6. An odd number is one which cannot be divided by 2 without a remainder ; as, 1, 3, 5, 7, 9, 15. OBS. All even numbers except 2, are composite numbers; an odd number is sometimes a composite, and sometimes a prime number. 7. One number is a measure of another, when the former is contained in the latter, any number of times without a remainder. Thus, 3 is a measure of 15 ; 7 is a measure of 28, &c. 8. One number is a multiple of another, when the former can be divided by the latter without a remainder. Thus, 6 is a multiple of 3 ; 20 is a multiple of 5, &c. OBS. A multiple is therefore a composite number, and the number thus contained in it, is always one of its factors. 9. The aliquot parts of a number, are the parts by which it can be measured or divided without a remainder. Thus, 5 and 7 are the aliquot parts of 35. 10. The reciprocal of a number is the quotient arising from dividing a unit by that number. Thus, the reciprocal of 2 is g; the reciprocal of 3 is }; that of 4 is , &c. 11. The difference between a given number and 10, 100, 1000, &c., that is, between the given number and the next higher order, is called the ARITHMETICAL COMPLEMENT of that number. Thus, 3 is the complement of 7; 15 is the complement of 85. Obs. The arithmetical complement of a number consisting of one integral figure, either with or without decimals, is found by subtracting the number from 10. If there are two integral figures, they are subtracted from 100; if three, from 1000, &c. 12. A perfect number is one which is equal to the sum of all its aliquot parts. Thus, 6=1+2+-3, the sum of its aliquot parts, and is a perfect number. Obs. 1. All the numbers known, to which this property really belongs, are the following: 6; 28; 496; 8128; 33,550,336; 8,589,869,056; 137,438,691,328; and 2,305,843,008,139,952,128.* 2. All perfect numbers terminate with 6, or 28. QUEST.-When is one number a measure of another ? What is a multiple? What are aliquot parts? What is the reciprocal of a number? * Hutton's Mathematical Recreations. 161. By the term properties of numbers, is meant those qualities or elements which are inherent and inseparable from them. Some of the more prominent are the following: 1. The sum of any two or more even numbers, is an even number. 2. The difference of any two even numbers, is an even number. 3. The sum or difference of two odd numbers, is even ; but the sum of three odd numbers, is odd. 4. The sum of any even number of odd numbers, is even ; but the sum of any odd number of odd numbers, is odd. 5. The sum, or difference, of an even and an odd number, is an odd number. 6. The product of an even and an odd number, or of two even numbers, is even. 7. If an even number be divisible by an odd number, the quotient is an even number. 8. The product of any number of factors, is even, if any one of then be even. 9. An odd number cannot be divided by an even number withvut a remainder. 10. The product of any two or more odd numbers, is an odd number. 11. If an odd number divides an even number, it will also divide the half of it. 12. If an even number be divisible by an odd number, it will also be divisible by double that number. 13. Any number that measures two others, must likewise measure their sum, their difference, and their product. 14. A number that measures another, must also measure its multiple, or its product by any whole number. 15. Any number expressed by the decimal notation, divided by 9, will leave the same remainder, as the sum of its figures or digits divided by 9. Demonstration.--Take any number, as 6357; now separating it into its several parts, it becomes 6000+300+50+7. But 6000=6X1000=6X(999+1) =6X999+6. In like manner 300=3X99+3, and 50=5X9+5. Hence 6357=6X999+3X99+5X946+3+5+7; and 6357-9=(6X999+3X99+ QUEST.-161. What is meant by properties of numbers ? |