The remainder when $7^{84}$ is divided by $342$ is

### Concept & Strategy When dividing a huge exponent by a specific number, the goal is to rewrite the base as a power that is exactly $1$ more or $1$ less than the divisor (or a multiple of the divisor). This allows us to use the Remainder Theorem: $(a \cdot x \pm 1)^n \pmod x \equiv (\pm 1)^n$. ### Step-by-Step Solution * **Given:** Find the remainder for $7^{84} \div 342$. * **Find a strategic base:** We need a power of $7$ that is close to $342$. Let us test the first few powers of $7$: $7^1 = 7$ $7^2 = 49$ $7^3 = 343$ * **Rewrite the expression:** Notice that $343$ is exactly $1$ more than the divisor $342$. We must restructure $7^{84}$ to use $7^3$. Divide the exponent $84$ by $3$. $84 \div 3 = 28$ $$7^{84} = (7^3)^{28} = 343^{28}$$ * **Apply modular arithmetic:** Now divide $343^{28}$ by $342$. $$343 \equiv 1 \pmod{342}$$ $$1^{28} = 1$$ ### Exam Strategy & Shortcut Memorizing the powers of single-digit numbers up to their cubes is crucial. If you know $7^3 = 343$, you immediately see the $+1$ relationship with $342$. Because $343$ leaves a remainder of $1$, and $1$ raised to *any* power is $1$, you don't even need to calculate the new exponent ($28$). The answer must be $1$. ### Common Pitfall Students who don't know $7^3 = 343$ will try to find a repeating cycle for the units digit of $7$, which does not help when dividing by a large three-digit number like $342$. ### Final Answer Therefore, the correct answer is **1**.