Find the values of x which will make each of the following expressions a perfect square:

15. x+6x8 + 13 x2 + 13 x 1. 16. x+6x + 11 x2 + 3x + 31. 17. x4 · 2 ax3 + (a2 + 2 b)x2 − 3 abx + 2 b2.

Find the values of x which will make each of the following expressions a perfect cube :

24. If n be any positive integer, prove that 52n-1 is always divisible by 24.

Find the factors of

25. a(b −c)3 + b(c − a)3 + c(a — b)3.

26. a(b ~ c)2 + b (c − a)2 + c(a − b)2 + 8 abc.

CHAPTER XXXI.

INDETERMINATE EQUATIONS OF THE FIRST DEGREE.

321. In Art. 167 we saw that if the number of unknown quantities is greater than the number of independent equations, there will be an unlimited number of solutions, and the equations will be indeterminate. By introducing conditions, however, we can limit the number of solutions. When positive integral values of the unknown quantities are required, the equations are called simple indeterminate equations.

The introduction of this restriction enables us to express the solutions in a very simple form.

Ex. 1. Solve 7 x + 12 y = 220 in positive integers.
Transpose and divide by the smaller coefficient; thus,

Since x and y are to be integers, we must have

Now multiplying the numerator by such a number that the division of the coefficient of y may give a remainder of unity, in this case 3, we have

that is,

and therefore

9-15y =

7

integer;

Substituting this value in the original equation, we obtain

Equation (1) shows that m may be 0 or have any negative integral value, but cannot have a positive integral value.

Equation (2) shows in addition that m may be 0, but cannot have a negative integral value greater than 2. Thus the only positive integral values of x and y are obtained by placing m = = 0, −1, −2. The complete solution may be exhibited as follows:

m = 0, −1, −2,

x = 28, 16, 4,

y = 2, 9, 16.

=

Ex. 2. Solve 5x-14 y 11 in positive integers
Proceeding as in Example 1, we obtain

(1).

This is called the general solution of the equation, and by giving to m any positive integral value, we obtain an unlimited number of values for x and y: thus we have

From Examples 1 and 2 the student will see that there is a further limitation to the number of solutions according as the terms of the original equations are connected by + or. If we have two equations involving three unknown. quantities, we can easily combine them so as to eliminate one of the unknown quantities, and can then proceed as above.

or

Ex. 3. In how many ways can $5 be paid in quarters and dimes? Let x the number of quarters, y the number of dimes; then

and

.. x = 2p,

y = 50 — 5 p.

Solutions are obtained by giving to p the values 1, 2, 3, ..., 9; and therefore the number of ways is 9. If, however, the sum be paid either in quarters or dimes, p may also have the values 0 and 10. If p = 0, then x = 0, and the sum is paid entirely in dimes; if p = 10, then y = 0, and the sum is paid entirely in quarters. Thus if zero values of x and y are admissible, the number of ways is 11.

EXAMPLES XXXI.

Solve in positive integers:

1. 3x+8y = 103. 3. 7 x + 12 y = 152.

2. 5x+2y

5. 23x+25 y = 915.

53. 4. 13x + 11 y=414. 6. 41 x + 47 y = 2191.

Find the general solution in positive integers, and the least values of x and y which satisfy the equations:

-

11. 19 y
12. 77 y

23 x

= 7.

7. 5x-7y = 3. 9. 8x21 y = 33. 8. 6x13y = 1. 10. 17 y — 13 x = 0. 30 x = 295. 13. A farmer spends $752 in buying horses and cows; if each horse costs $37, and each cow $23, how many of each does he buy?

14. In how many ways can $100 be paid in dollars and half-dollars, including zero solutions?

15. Find a number which, being divided by 39, gives a remainder 16, and, by 56, a remainder 27. How many such numbers are there?

CHAPTER XXXII.

INEQUALITIES.

-

322. Any quantity a is said to be greater than another quantity b when ab is positive; thus 2 is greater than -3, because 2 -(-3), or 5, is positive. Also b is said to be less than a when b a is negative; thus 5 is less than -2, because-5-(-2), or -3, is negative. In accordance with this definition, zero must be regarded as greater than any negative quantity.

323. The statement in algebraic language that one expression is greater or less than another is called an inequality.

324. The sign of inequality is >, the opening being placed towards the greater quantity. Thus, a >b is read "a is greater than b.”

325. The first and second members are the expressions on the left and right, respectively, of the sign of inequality.

326. Inequalities subsist in the same sense when corresponding members in each are the greater or the less. Thus, the inequalities a > b and 7 >5 are said to subsist in the

same sense.

In the present chapter, we shall suppose (unless the contrary is directly stated) that the letters always denote real and positive quantities.

327. Inequality Unchanged. An inequality will still hold after each side has been increased, diminished, multiplied, or divided by the same positive quantity.