Key to Robinson's University Algebra: Containing, Also, a Short Treatise on the Indeterminate and Diophantine Analysis. And Some Miscellaneous Examples

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Ivison, Phinney, 1863 - Algebra - 128 pages
 

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Page 92 - And from equation (2), 2= — 2-|-y, or #=4, and lly==44, one of the numbers, and of course 56 is the other, 8. Find a number which being divided by 6, shall leave the remainder 2, and the same number divided by 13 shall leave the remainder 3. Consider that in division, the divisor and quotient multiplied together, and the remainder added, gives the number divided. Let N represent the number divided, x and y the quotients. Then 6*+2=N, and l3y+3=N.
Page 113 - H, 1, are the numbers ; but we can multiply them all by the same square number 64, and their arithmetical relation will not be changed, and they will still be squares ; hence 1, 25, and 49 may be the numbers, or 4, 100, and 196, 5. Find two whole numbers, such that the sum and difference of their squares, when diminished by unity, shall be a square. Let a:-{-l=one number, and y= the other.
Page 110 - The preceding are some of the most comprehensive and general methods yet known ; but there are cases in practice where no general rules will be so effectual, as the operator's own judgment and penetration. Much, very much will depend on skill and foresight displayed at the commencement of a problem, by assuming convenient expressions to satisfy one or two conditions at once, and the remaining conditions can be satisfied by some one of the preceding rules. EXAMPLES. 1. It is required to find three...
Page 114 - Find two numbers whose product is equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes.
Page 123 - He finds that if he buys 120 acres of cleared land, and lays out the rest of his money for that which is not cleared, he will not get the quantity of land he wants by 25 acres, but, if he buys 220 acres of uncleared land, and then buys a sufficient number of acres of cleared land to make up the number of acres he wants, he will have 4 dollars left. How many acres of each must he buy to have the quantity he wishes, and lay out all his money? (Harney page 203. Ans. 20 acres cleared, 218 uncleared....
Page 113 - To find three whole numbers such, that if to the square of each the product of the other two be added, the three sums shall be all squares. Ans. 9, 73, and 328.
Page 126 - The sum of 700 dollars was divided among four persons. A, B, C, and D, whose shares were in geometrical progression ; and the difference between the greatest and' least, was to the difference between the two means, as 37 to 12. What were the several shares? 19. The sum of three numbers in harmonical proportion is 191, and the product of the first and third is 4032 ; required the numbers. 20. The 2d and 6th terms of a geometrical progression are respectively 21 and 1701. What is the first term, and...
Page 126 - AnS\0,16l25) 9. A person has 27 6s. in guineas and crown pieces, out of which he pays a debt of 14 17s., and finds that he has exactly as many guineas left as he has paid...
Page 126 - Is it possible to pay 50 by means of guineas and three shilling pieces only. Ans. Impossible. 12. A merchant drew every year upon the stock he had in trade, the sum of a dollars for the expense of his family. His profits each year, were the...
Page 115 - Find three integral square numbers in harmonical proportion. Ans. 25, 49, and 1235. 14. Find two numbers in the proportion of 8 to 15, and such that the sum of their squares shall be a square number. Ans. 136 and 255.

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