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Signs Do Not Affect Parity

Signs Do Not Affect Parity

Idea · Beginner

Number Theory

\left||a|-|b|+|c|-|d|\right| \equiv a+b+c+d \pmod{2}

Modular Arithmetic

For any integer $a$ ,

a-(-a)=2a

Since $2a$ is even,

a\equiv -a\pmod{2}

Thus changing the sign of any term does not change the parity of a sum.

How many ways are there to select $+$ and $-$ to satisfy

\pm1\pm2\pm\cdots\pm100=101

\begin{aligned} \pm1\pm2\pm\cdots\pm100 &\equiv 1+2+\cdots+100\\ &= \frac{100\cdot101}{2}\\ &= 50\cdot101\\ &\equiv0\pmod2 \end{aligned}

So the left-hand side is always even, so it cannot be equal to 101.

\boxed{0}

Prove that

|a-b|+|b-c|+|c-a|

is even.

\begin{aligned} |a-b|+|b-c|+|c-a| &\equiv a-b+b-c+c-a\\ &\equiv0 \pmod2 \end{aligned}

AIME I 2025 (Problem 5)

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