The square of a number greater than 1000 that is not divisible by three, when…

Key idea: any integer is congruent to 0, 1, or 2 modulo 3, and for numbers not divisible by 3 the square is always congruent to 1 modulo 3.

• If the number is congruent to 1 modulo 3, then its square is 1^2 = 1 modulo 3.

• If the number is congruent to 2 modulo 3, then its square is 2^2 = 4 ≡ 1 modulo 3.

• Since the problem excludes numbers divisible by 3, the original number must be congruent to 1 or 2 modulo 3, and in both cases the square is congruent to 1 modulo 3.

Conclusion: The square of any number greater than 1000 that is not divisible by 3 leaves a remainder of 1 when divided by 3.

Example: 1001 ≡ 2 (mod 3), and 1001^2 ≡ 4 ≡ 1 (mod 3), so the remainder is 1.