<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-remainders-and-exponents">Remainders and Exponents Exercise</a></p><p>A GRE problem might ask you to solve a <strong><span style="color:#27ae60;">Remainder and Exponents</span></strong> problem, where a very large exponent is in play. You can see three examples below:</p>
<p><span style="font-size:20px;"><span style="color:#e74c3c;">Examples</span></span></p>
<ul>
<li>What is the remainder of $7^{37} \div 3$?</li>
<li>What is the remainder of $5^{50} \div 6$?</li>
<li>What is the remainder of $13^{20} \div 10$?</li>
</ul>
<p><span style="font-size:20px;"><span style="color:#e74c3c;">How to Solve</span></span></p>
<p>It depends on what you're dividing by:</p>
<ul>
<li>$\div \ 1$: The remainder is always $0$ becuase every integer is divisible by $1$.</li>
<li>$\div \ 2$: The remainder is either $0$ (if the number being divided is even) or $1$ (if the number being divided is odd).</li>
<li>$\div \ 3$: Write down the first one to five remainders and see if you can find some kind of pattern. For example, what is the remainder of $7^{35} \div 3$?
<ul>
<li>Remainder of $7^1 \div 3 = 1$</li>
<li>Remainder of $7^2 \div 3 = 1$</li>
<li>Remainder of $7^3 \div 3 = 1$</li>
<li>Ahh, so we can see the remainder must be $1$.</li>
</ul>
</li>
<li>$\div \ 4$: Calculate the final two digits of the number being divided and use your divisibility rule with $4$ to determine the remainder.</li>
<li>$\div \ 5$: Calculate the unit digit of the number being divided. If the unit digit $0$, $1$, $2$, $3$, or $4$, that's the remainder. If the unit digit is $5$, $6$, $7$, $8$, or $9$, the remainder is equal to the unit digit minus $5$.</li>
<li>$\div \ 6$: Try to find some kind of pattern (like in the $\div \ 3$ case).</li>
<li>$\div \ 7$: Try to find some kind of pattern (like in the $\div \ 3$ case).</li>
<li>$\div \ 8$: Calculate the final three digits of the number being divided and use your divisibility rule with $8$ to determine the remainder.</li>
<li>$\div \ 9$: Try to find some kind of pattern (like in the $\div \ 3$ case).</li>
<li>$\div \ 10$: Simply calculate the unit digit of the number being divided. That's the remainder.</li>
</ul>