By Long Luo

# Math

If $n$ is a power of $4$, it must be $n = 4^x, x \ge 0$.

Then:

$4^x \equiv (3+1)^x \equiv 1^x \equiv 1 \quad (\bmod ~3)$

If $n = 2^x$ but $n \ne 4^x$, it must be $n = 4^x \times 2$, which means $n \equiv 2 \quad (\bmod ~3)$.

Therefore, we can check whether $n = 4^x$ by whether $n \equiv 1 \quad(\bmod ~3)$.

## Analysis

• Time Complexity: $O(1)$.
• Space Complexity: $O(1)$.

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