The digit in the unit's place of the number $(67)^{25} - 1$ must be

### Concept & Logic To find the unit digit of an expression involving subtraction, calculate the unit digit of the exponentiation first using power cyclicity, and then perform the subtraction on that unit digit. ### Step-by-Step Solution **Given:** The expression is $(67)^{25} - 1$. **Calculation:** * First, isolate the exponentiation part: $(67)^{25}$. We only care about the unit digit of the base, which is $7$. * The cyclicity of $7$ is $4$ (the pattern is $7, 9, 3, 1$). * Find the remainder of the exponent divided by the cyclicity: $$25 \pmod 4 = 1$$ * A remainder of $1$ means the unit digit corresponds to the $1$st position in the cycle, which is equivalent to $7^1 = 7$. * Now, substitute the unit digit back into the original expression: $$7 - 1 = 6$$ ### Exam Strategy & Shortcut Exponent $25$ is just $1$ more than $24$ (a multiple of $4$). So the power of $7$ resets to the start of its cycle, ending in $7$. Subtract $1$ from $7$ to instantly get $6$. This should take under $5$ seconds mentally. ### Common Pitfall A common error is subtracting the $1$ from the base *before* applying the exponent (i.e., treating it as $66^{25}$), which is mathematically incorrect according to the order of operations and leads to a completely wrong unit digit. ### Final Answer **Therefore, the correct answer is 6.**