a, b, and x are positive integers such that
sqrt(a) + sqrt(b) = sqrt(x)
How many possible values of x less than or equal to 1000 are there?
(In reply to
re(2): First thoughts... ...(spoiler?) by Erik O.)
1st: there's no reason why a, b and x need to be perfect squares. sqrt(2) + sqrt(18) = sqrt(32)
2nd: not every pair (a,b) should be counted, as some pairs have square roots that add to the same number, such as sqrt(1)+sqrt(25)=sqrt(4)+sqrt(15)=sqrt(9)+sqrt(9)=sqrt(36), so that's only one value of x, i.e., 36.
3rd: when counting only perfect squares, I count only 30 values of x, through 240 possible sums (not counting reversals).
Counting values that are not perfect squares, but not counting reversals of a and b, I get 355 different values of x through 904 possible sums. So I think the answer is 355.
The solutions that are perfect squares are:
1 1 4
1 4 9
1 9 16
4 4 16
1 16 25
4 9 25
1 25 36
4 16 36
9 9 36
1 36 49
4 25 49
9 16 49
1 49 64
4 36 64
9 25 64
16 16 64
1 64 81
4 49 81
9 36 81
16 25 81
1 81 100
4 64 100
9 49 100
16 36 100
25 25 100
1 100 121
4 81 121
9 64 121
16 49 121
25 36 121
1 121 144
4 100 144
9 81 144
16 64 144
25 49 144
36 36 144
1 144 169
4 121 169
9 100 169
16 81 169
25 64 169
36 49 169
1 169 196
4 144 196
9 121 196
16 100 196
25 81 196
36 64 196
49 49 196
1 196 225
4 169 225
9 144 225
16 121 225
25 100 225
36 81 225
49 64 225
1 225 256
4 196 256
9 169 256
16 144 256
25 121 256
36 100 256
49 81 256
64 64 256
1 256 289
4 225 289
9 196 289
16 169 289
25 144 289
36 121 289
49 100 289
64 81 289
1 289 324
4 256 324
9 225 324
16 196 324
25 169 324
36 144 324
49 121 324
64 100 324
81 81 324
1 324 361
4 289 361
9 256 361
16 225 361
25 196 361
36 169 361
49 144 361
64 121 361
81 100 361
1 361 400
4 324 400
9 289 400
16 256 400
25 225 400
36 196 400
49 169 400
64 144 400
81 121 400
100 100 400
1 400 441
4 361 441
9 324 441
16 289 441
25 256 441
36 225 441
49 196 441
64 169 441
81 144 441
100 121 441
1 441 484
4 400 484
9 361 484
16 324 484
25 289 484
36 256 484
49 225 484
64 196 484
81 169 484
100 144 484
121 121 484
1 484 529
4 441 529
9 400 529
16 361 529
25 324 529
36 289 529
49 256 529
64 225 529
81 196 529
100 169 529
121 144 529
1 529 576
4 484 576
9 441 576
16 400 576
25 361 576
36 324 576
49 289 576
64 256 576
81 225 576
100 196 576
121 169 576
144 144 576
1 576 625
4 529 625
9 484 625
16 441 625
25 400 625
36 361 625
49 324 625
64 289 625
81 256 625
100 225 625
121 196 625
144 169 625
1 625 676
4 576 676
9 529 676
16 484 676
25 441 676
36 400 676
49 361 676
64 324 676
81 289 676
100 256 676
121 225 676
144 196 676
169 169 676
1 676 729
4 625 729
9 576 729
16 529 729
25 484 729
36 441 729
49 400 729
64 361 729
81 324 729
100 289 729
121 256 729
144 225 729
169 196 729
1 729 784
4 676 784
9 625 784
16 576 784
25 529 784
36 484 784
49 441 784
64 400 784
81 361 784
100 324 784
121 289 784
144 256 784
169 225 784
196 196 784
1 784 841
4 729 841
9 676 841
16 625 841
25 576 841
36 529 841
49 484 841
64 441 841
81 400 841
100 361 841
121 324 841
144 289 841
169 256 841
196 225 841
1 841 900
4 784 900
9 729 900
16 676 900
25 625 900
36 576 900
49 529 900
64 484 900
81 441 900
100 400 900
121 361 900
144 324 900
169 289 900
196 256 900
225 225 900
1 900 961
4 841 961
9 784 961
16 729 961
25 676 961
36 625 961
49 576 961
64 529 961
81 484 961
100 441 961
121 400 961
144 361 961
169 324 961
196 289 961
225 256 961
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Posted by Charlie
on 2005-05-05 19:47:04 |
re(3): First thoughts... ...(spoiler?)
