QUESTIONS IN THE DIOPHANTINE ANALYSIS. 219

6. What numbers divided by 6, 7 and 8, give the remainders 5, 6 and 7?

7. A man wishes to divide 100 dollars between 4 men and 5 women; so he divided the 100 dollars into two parcels, giving one parcel to the men, and the other to the women. How many dollars were there in each parcel? and how many different ways could he divide it?

8. A man wishes to purchase some sugar and coffee for 159 cents; the sugar at 14 cents a pound, and the coffee at 15. How many pounds of each can he buy, to lay out all his money?

9. Divide 150 into three such parts, that if the first be multiplied by 2, the second by 3, and the third by 5, the sum of the products may be 530.

10. Find two numbers whose sum and product are equal.

11. Find two numbers whose difference is equal to the difference of their second powers.

12. A man bought 6 pounds of sugar and 5 pounds of coffee for 115 cents. How much do the sugar and coffee cost per pound, that they may be as near the same price as possible, and cost, each, a whole number of cents?

SECTION XCII.

Questions in the Diophantine Analysis.

1. Find a value of x, such that the following expressions shall be square numbers:

2. Find two square numbers whose difference is 4.

3. Find what will represent two square numbers, whose difference is b.

4. Find two square numbers whose sum is 625.

5. Find two square numbers whose sum is a2.

6. Find a value of x, such that x+28 and x + 30, may be both squares.

7. Find two numbers, such that their difference shall be equal to the difference of their squares, and the sum of their squares a square number.

8. Find two whole numbers, such that their sum and difference shall be both squares.

9. Divide the number 100 into three square numbers.

10. Find three numbers in geometrical progression, such that the sum of the three shall be a square number.

11. Will the following expressions always represent square numbers, whatever numbers the letters represent? and what will their roots be?

12. Find two numbers, such that the sum of their squares shall be a square,

p2-q2 and 2pq will always represent such numbers, whatever values we assign to p and 9. Why?

Any number divided by 2 must give a remainder, either O

or 1.

13. Prove that any square number, divided by 3, must give 0 or 1 for a remainder.

Let

Now

x2= the square number.

must give the remainder 0, 1 or 2, and we have

813

Hence gives for a whole number and; consequently a

3

whole number, and 1 for a remainder.

14. Prove, that if a square number be divided by 4, it gives 0 or 1 for a remainder. By 5; 0 or 1, 4. By 6; 0, 1, 3, 4. By 7; 0, 1, 2, 4. By 8; 0, 1, 4. By 9; 0, 1, 4, 7. By 10; 0, 1, 4, 5, 6, 9.

a

15. Show that 3x+2 can never be a perfect, square, if "whole numbers be assumed for x.

16. Prove that 5x+2, or 5x + 3 can never be perfect ́squares, if x be a whole number.

17. Can 8x +2, 8 x2 + 3, or 8x+7, ever be square numbers, if x be a whole number?

18. Find the product of two factors which shall be equal to the sum of the squares of two square numbers, also found.

Assume p2+q2 and p2 + q2 for the two factors.

Then (p+q) (p + q2) = (pp' + q q')2 + (p q' — qp')' we may then assume p and q any numbers whatever.

19. Find three numbers, such that if their sum be added to, or subtracted from, the square of each of these, the sum and difference shall be each squares.

20. Divide a given cube number into three other cube

numbers

SECTION XCIII..

On Annuities.

1. Suppose a man lends 100 dollars for one year, at 6 per cent. per annum; what will the interest be? What will the amount be?

2. Suppose he lends this amount the second year, at the same rate; what will the interest be? What will the amount be?

3. Suppose a man proceeds in this way, lending each year, the amount of the preceding year; what will the amount be at the end of 5 years?

When money is loaned, as supposed in the preceding questions, it is said to be loaned at compound interest.

24. In the equation p = a, what is the value of r? First divide by p, and we have

r is the amount of 1 dollar for 1 year. Subtract 1 from it, and the remainder is the interest of 1 dollar for 1 year, or what is called the rate per unit. One hundred times the in

terest of 1 dollar, or the rate per unit, gives the interest of 100 dollars, or the rate per cent.?

25. Sixty dollars amount to 70 in 3 years. What is the rate per cent.?

26. One hundred and fifty dollars amount to 165 dollars in 11 years. What is the rate per cent.?

27. One hundred and thirty dollars amount to 140 in 3 years. What is the rate per cent.?

28. In the equation pr2 = a, what is the value of n? Divide both members by p, and we have

Take the logarithms of both members, and we have

Divide the difference between the logarithm of the amount and principal, by the logarithm of r, the amount of 1 for 1 year. The quotient is the number of years.

29. In what time will 150 amount to 165, at 8 per cent.? 30. In what time will 35 dollars amount to 40, at 5 per cent.?

31. In what time will any principal be doubled at 6 per cent.?

When the principal is doubled, the amount a is 2p, and the equation pr=a becomes pr=2p. Dividing by p, we

have

32. In what time will any principal be tripled, at 4 per cent.?

33. In what time will any principal be quadrupled at 8 per

cent?

SECTION XCIV.

On Annuities.

1. Suppose a man deposits annually 10 dollars, in a savings bank, for 12 years; what sum is he entitled to draw out, supposing he is allowed 4 per cent. compound interest on each deposit?

Letr the amount of 1 for 1 year, which in the preceding question is 1.04. The 10 dollars deposited the first year is at interest for the whole time; and its amount is 10 ₪12. The 10 deposited the second year, is at interest 11 years; and its amount is 10 μll. The 10 deposited the third, is at interest 10 years; and its amount is 1010, &c.

The amounts on each deposit make the following series of

terms:

10 r12, 10 r11, 10 r1o, 10 r3, 10 r3, 10 r7, 10 ro, 10 r3, 10 ra, 10 r3, 10 r2, 10 r.

Number these terms from right to left. It will be seen that this series is a geometrical progression; for each term multiplied by r, gives the succeeding one. The sum of all these terms gives the whole amount, or the sum that the man is entitled to draw at the end of 12 years; but, the sum of this