16 5x by = 13 and xy = 6, then find the value of
125 X3 - 216 y 3,

For example, the system x 2 - 2u2 = 2, y2 - 2v2 = -16 along with the given

system is satisfied by

(x, y, u, v) = (2, -4,1,4), (10,4,7,4), (2, -28, -1,20).

Note that (lxi, Iyl, lui, Ivl) = (10,28,7,20) also satisfies the pair ofPell's equations, but no adjustment of signs of the entries will yield a solution of the original

system.

The pair x 2 - 2u2 = ±2r, y2 - 2v2 = =t=2s will arise from any system

2uv - xy = a, xv - uy = b for which a2 - 2b2 = -32. Each of a and b must be

divisible by 4; possible values are given by (Ial, Ibl) = (0,4), (16, 12), (96,68).

2.7. 2x2 + (x ± 1)2 = w2 can be rewritten as (3x ± 1)2 - 3w2 = -2. Some

solutions are (3x ± 1, w) = (1,1), (5, 3), (19,11), (71,41), (265, 153).

Step-by-step explanation:

452,489 to the nearest thousand is 452,000

37

step-by-step explanation: