Consider the remainders obtained when the given 181 square integers are divided by 19. If n is any integer then n is congruent modulo 19
to one of the numbers 0, 1, 2, . . ., 18 and hence n^2 is congruent to 0, 1, 4, 9,16, 6, 17, 11, 7 or 5.
Hence there are only 10 possible values for the remainders. Since there are 181 remainders at least, one of them must repeat 19 times at least. Choose 19 numbers congruent to that remainder. This
proves the result.
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