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And x=69308+4387=(if s=0) 4387.

Assuming s=1, 2, &c. other values would arise.

2. Required a number which being divided by 2, 3, 4, 6, shall leave no remainder; but being divided by 7, the remainder shall be 6?

The least common multiple of 2, 3, 4, 6,* is 12.

Let, therefore, 12x= the number sought.

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-4 =wh; and 2==wh=p. ..x=7p+4.

And 12x=84p+48=48, 132, 216, &c.

3. Required the least whole number, which being divided by 3, 5, 7, and 8, shall leave the remainders 1, 3, 5, and Ŏ, respectively?

Ans. 208.

4. Required the least whole number, which being divided by 9, 10, 11, and 12, shall leave the remainders 4, 5, 6, and 7, respectively? Ans. 1975.

5. What is the least whole number, which being divided by each of the nine digits shall leave no remainder; but divided by 17 the remainder shall be 15?

Ans. 40320.

6. In what year of the Christian era, was the solar cycle 15, the lunar cycle 3, and the Roman indiction 14计

Ans. 1826.

*

See Arithmetical Expositor, Part I, page 73.

The solar cycle is a period of 28 years, at the end of which, in case a centurial year has not intervened, the days of the week always return to the same days of the month. To the year of

7. Required the least whole number, which being divided by 2, 3, 4, and 5, shall leave a remainder of 1; but divided by 7, there shall nothing remain.

Ans. 301.

MISCELLANEOUS EXAMPLES.

130.-1. A bankrupt owes A. twice as much as he owes B., and he owes to C. as much as to A. and B. together; the sum to be divided is 600 dollars; how much must each receive?

Ans. A. 200, B. 100, C. 300.

2. A. can perform a piece of work in 7 days, and B. in 9 days; in what time would they jointly effect it? Ans. 31 days.

3. A labourer being employed on condition, that for every day he worked he should receive 50 cents, and for every day he was idle, he should forfeit 20 cents, finds at the end of 500 days, only 89 dollars due; how many days did he work?

Ans. 290.

4. There are two numbers in the ratio of 4 to 5, and the sum of their squares is 1476; what are the numbers? Ans. 24 and 30.

5. A Greek epitaph, designed for the tomb of Diophantus, is said to have stated that he passed one sixth of his life in childhood; one twelfth in adolescence; that after one seventh and five years more had been passed

the Christian era, add 9, and divide by 28, the remainder will be the number of the cycle.

The lunar cycle is a period of 19 years, at the expiration of which, the new and full moons return nearly to the same time of the year. To the year of the Christian era add 1, and divide by 19, the remainder is the lunar cycle.

The Roman indiction is not an astronomic period, it consists of 15 years. To a given year add 3, and divide by 15, the remainder is the number of the indiction.

If, in either of these cases, no remainder occurs, the divisor must be taken as the number of the cycle.

in a married state, he had a son, who lived to half his own age, and whom he survived four years; what then was the age of Diophantus ? Ans. 84 years.

6. A person's age in years is a number, consisting of two digits; of this number is a mean proportional bebetween these digits; and two years hence his age will be a third proportional to those digits, beginning with the tens; what is the age? Ans. 14 years.

7. Given x+y+z=9, xy+xz+yz=26, xyz=24, to find the values of x, y, and z.

Result, x=2, 3, or 4.

8. By selling a piece of muslin at a certain price per yard, I gained the prime cost of 9 yards, which was just as much per cent. as the number of yards in the piece; Ans. 30 yards.

what was that number?

9. There are 4 numbers in continued proportion, the sum of the first and last 1728, and the sum of the other two 1152; what are the numbers ?

Ans. 192, 384, 768, and 1536.

10. The hypothenuse of a right angled triangle is x3x, and the other sides x2, and x; what is the area?

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11. There is a number consisting of three digits in arithmetical progression, which number being divided by the sum of the digits the quotient will be 48, but if from the number 198 be subtracted, the remainder will be expressed by the same digits in an inverted order. Quere the number? Ans. 432.

12. The sum of the squares of two numbers being multiplied by the quotient arising from the division of the less by the greater, produces 85.2, and the difference of the squares multiplied by the quotient of the greater divided by the less, produces 1920; what are the numbers ? Ans. 20 and 4.

13. A ball falling from the top of a tower, is observed

to descend one fourth of the distance in the last second of the time; required the height of the tower, heavy bodies being known to fall 16 feet during the first second, and to describe spaces, which, reckoned from the beginning of the fall, are as the squares of the times ? Ans. (28±16/3)16 feet,

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14. What are the values of x and z, from the equations x8z+xz3=546560, xa+z+=1086992 ?

Ans. x=32, z=14.

15. Given x+y+z+v=56, x2+y2+z2+v2=910, xv+2y2--z-6, and z=2y, to find the values of x, y, z, and v. Result, x=8, y=9, z=18, v=21.

16. Required to find a number, which being any way divided into two unequal parts, the greater part added to the square of the less, shall be equal to the less part added to the square of the greater ? Ans. 1.

17. Given xy=125x+300y, and y3-x2-90000, to find x and y by a quadratic.

Result, x=400, y=500.

18. Given (x+1), (x2+1), (x+1)=30x3, to find the value of x, by a quadratic equation.

Result, x=1(3±√5.)

19. The sum of three numbers in harmonical proportion is 26, and their continued product 576; required the numbers by a quadratic equation?

Ans. 12, 8, and 6.

20. The sum of two numbers is 152, and the cube root of the square of their difference multiplied by the square root of the cube of the same difference, produces 8192; what are the numbers? Ans. 44 and 108.

21. The sum of two numbers added to the sum of their squares is 120, and the product of the same numbers 45; what are they? Ans. 9 and 5.

22. Given 2+xy=4640y, and xy-y3-537.6.x, what are the numerical values of x and y?

Ans. x=40, y=16.

23. Given x+y=z, 23⁄4—x3+y3=470, z3—x3—y3 — 468, to find x, y, and z.

Result, x=12, y=1, z=13.

24. Given v2+vx+vy+vz=252, x2+xv+xy+xz= 504, y2+yv+yx+yz=396, zo+zv+zy+zx=144, to find v, x, y, and z.

Result, v=7, x=14, y=11, z=4.

.

25. Given x-2x+x=a, to find x by a quadratic equation. Result, x=1+√($+√@+4.)

26. Given x2+(y+z)=a, y3+y(x+z)=b, z2+z (x+y)=c, to find x, y, and z.

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Result, x=5, y=20.

27. Given 2x3—x1—x2 + 2x2y—y2=2xy=√✅✅x3y—

xy, to find x and y.

28. Required the roots of the equation 4x+8289x2+28x+49=0, by quadratics only?

7 13+113 -13-113

Ans. 1,

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29. There are three numbers in geometrical progression, whose continued product is 4096, and the sum of the extremes is 68; what are the numbers?

Ans. 4, 16, and 64.

30. There are two numbers expressed by the same two digits, and the difference of their squares is 1485; quere the numbers ?

Ans. 14 and 41.

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