If $7^{126}$ is divided by $48$, find the remainder.

### Concept & Strategy When finding remainders of large powers, the goal is to rewrite the base so it is exactly $1$ more or $1$ less than a multiple of the divisor. This allows you to use $1$ or $-1$ as the remainder, making the large exponent trivial to calculate. ### Step-by-Step Solution **Given:** Expression: $7^{126}$ Divisor: $48$ **Calculation / Deduction:** * Look for a power of $7$ that is close to a multiple of $48$. We know $7^1 = 7$, and $7^2 = 49$. * Notice that $49$ is exactly $1$ more than the divisor $48$. * Rewrite the original expression in terms of $7^2$: $$ 7^{126} = (7^2)^{63} = (49)^{63} $$ * Now, apply the modular arithmetic. When $49$ is divided by $48$, the remainder is $1$. $$ 49 \equiv 1 \pmod{48} $$ * Substitute this remainder back into the exponent: $$ 1^{63} = 1 $$ ### Exam Strategy & Shortcut Instantly check the squares or cubes of the given base against the divisor. You see $7$ and $48$. You should immediately associate $7^2 = 49$. Since $49$ is $1$ greater than $48$, it yields a remainder of $+1$. $+1$ raised to any power is always $1$. ### Common Pitfall Students often try to find the cyclicity of $7$ by dividing powers of $7$ by $48$ manually (e.g., finding $7^1/48$, $7^2/48$, $7^3/48$, etc.). This is a massive time sink. Always look to match the base to $(D+1)$ or $(D-1)$ where $D$ is the divisor. ### Final Answer Therefore, the correct answer is 1.