To prove that the square of any integer is either 0 or 1 modulo 4, consider the following: 1. Expressing Integers in Terms of Modulo 4: Any integer \( n \) can be written in the form \( 4k \), \( 4k + 1 \), \( 4k + 2 \), or \( 4k + 3 \), where \( k \) is an integer. 2. Squaring Each Form: - \( n = 4k \): \( (4k)^2 = 16k^2 \). When divided by 4, the remainder is 0. - \( n = 4k + 1 \): \( (4k + 1)^2 = 16k^2 + 8k + 1 \). The remainder is 1 after division by 4. - \( n = 4k + 2 \): \( (4k + 2)^2 = 16k^2 + 16k + 4 \). The remainder is 0 after division by 4. - \( n = 4k + 3 \): \( (4k + 3)^2 = 16k^2 + 24k + 9 \). The remainder is 1 after division by 4. 3. Conclusion: In all cases, the square of an integer modulo 4 yields either 0 or 1. Therefore, the square of any integer is either 0 or 1 modulo 4.

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To prove that the square of any integer is either 0 or 1 modulo 4, consider the following: 1. Expressing Integers in Terms of Modulo 4: Any integer \( n \) can be written in the form \( 4k \), \( 4k + 1 \), \( 4k + 2 \), or \( 4k + 3 \), where \( k \) is an integer. 2. Squaring Each Form: - \( n = 4k \): \( (4k)^2 = 16k^2 \). When divided by 4, the remainder is 0. - \( n = 4k + 1 \): \( (4k + 1)^2 = 16k^2 + 8k + 1 \). The remainder is 1 after division by 4. - \( n = 4k + 2 \): \( (4k + 2)^2 = 16k^2 + 16k + 4 \). The remainder is 0 after division by 4. - \( n = 4k + 3 \): \( (4k + 3)^2 = 16k^2 + 24k + 9 \). The remainder is 1 after division by 4. 3. Conclusion: In all cases, the square of an integer modulo 4 yields either 0 or 1. Therefore, the square of any integer is either 0 or 1 modulo 4.

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