Remainder

Find the remainder when 2 2048 + 3 ( 2 1025 ) + 9 \sqrt { { 2 }^{ 2048 }+3({ 2 }^{ 1025 })+9 } is divided by 6.

4 5 0 3 1 2

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1 solution

Chew-Seong Cheong
Jul 10, 2015

2 2048 + 3 ( 2 1025 ) + 9 ( 2 1024 ) 2 + 2 ( 2 1024 ˙ 3 ) + 3 2 ( 2 1024 + 3 ) 2 [ 2 1024 + 3 ] ( m o d 6 ) [ 4 + 3 ] ( m o d 6 ) [See Note.] 1 ( m o d 6 ) \begin{aligned} \sqrt{2^{2048} + 3(2^{1025}) + 9} & \equiv \sqrt{(2^{1024})^2 + 2(2^{1024}\dot{}3) + 3^2} \\ & \equiv \sqrt{(2^{1024}+3) ^2} \\ & \equiv [2^{1024}+3] \pmod{6} \\ & \equiv [\color{#3D99F6}{4} + 3] \pmod{6} \quad \quad \color{#3D99F6}{\text{[See Note.]}} \\ & \equiv \boxed{1} \pmod{6} \end{aligned}

Note: \color{#3D99F6}{\text{Note:}} From observations, we note that 2 n m o d 6 = { 2 when n is odd 4 when n is even \quad 2^n \mod{6} = \begin{cases} 2 & \text{when n is odd} \\ 4 & \text{when n is even} \end{cases}

Moderator note:

Good observation with the perfect square factorization.

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