Elements of Algebra

462 ELEMENTS OF ALGEBRA. PART II. QUESTIONS FOR PRACTICE. 1. To divide a square number (16) into two squares. Ans. $\frac{256}{25}$, and $\frac{144}{25}$. 2. To find two square numbers, whose difference (60) is given. Ans. $72\frac{1}{4}$, and $132\frac{1}{4}$. 3. From a number $x$ to take two given numbers 6 and 7, so that both remainders may be square numbers. Ans. $x = \frac{121}{16}$. 4. To find two numbers in proportion as 8 is to 15, and such, that the sum of their squares shall make a square number. Ans. 576, and 1080. 5. To find four numbers such, that if the square number 100 be added to the product of every two of them, the sum shall be all squares. Ans. 12, 32, 88, and 168. 6. To find two numbers, whose difference shall be equal to the difference of their squares, and the sum of their squares a square number. Ans. $\frac{4}{7}$, and $\frac{3}{7}$. 7. To find two numbers, whose product being added to the sum of their squares, shall make a square number. Ans. 5 and 3, 8 and 7, 16 and 5, \ldots 8. To find two such numbers, that not only each number, but also their sum and their difference, being increased by unity, shall be square numbers. Ans. 3024, and 5624. 9. To find three square numbers such, that the sum of their squares shall be a square number. Ans. 9, 16, and $\frac{144}{25}$. 10. To divide the cube number 8 into three other cube numbers. Ans. $\frac{64}{27}$, $\frac{125}{27}$, and 1. 11. Two cube numbers, 8 and 1, being given, to find two other cube numbers, whose difference shall be equal to the sum of the given cubes. Ans. $\frac{8000}{343}$, and $\frac{4913}{343}$. 12. To find three such cube numbers, that if 1 be subtracted from every one of them, the sum of the remainders shall be a square. Ans. $\frac{4913}{3375}$, $\frac{21952}{3375}$, and 8. 13. To find two numbers, whose sum shall be equal to the sum of their cubes. Ans. $\frac{5}{7}$, and $\frac{8}{7}$. 14. To find three such cube numbers, that the sum of them may be both a square and a cube. Ans. 1, $\frac{2084383}{374625}$, $\frac{15252992}{274625}$.