If you raise 3^100, what will be the units digit of the answer?

When you multiply a number by itself repeatedly, the last digit of the results can fall into a predictable, repeating pattern. For the number 3, this cycle is easy to observe. The answer to 3^1 ends in 3. For 3^2, the final digit is 9. For 3^3 (or 27), it's 7, and for 3^4 (or 81), it's 1. If we go to the next power, 3^5, the result is 243, and the final digit is 3 once again. This establishes a reliable four-number pattern for the units digit: 3, 9, 7, 1. This concept is often referred to as cyclicity.

To solve for a large exponent like 100, we simply need to determine where it falls within this four-digit sequence. Since the cycle has four digits, we can divide the exponent of 100 by 4. The result is a clean 25 with a remainder of 0. In this system, a remainder of 1 would point to the first number in the cycle (3), a remainder of 2 to the second (9), and so on. A remainder of 0 indicates that we have completed a full cycle, landing on the fourth and final digit in the pattern, which is 1.

This method of using remainders and cycles is a practical application of a branch of mathematics known as modular arithmetic. Sometimes called "clock arithmetic," it involves numbers that wrap around upon reaching a specific value, much like the hours on a clock face. The modern approach to this field was systemized by the renowned mathematician Carl Friedrich Gauss in his 1801 book "Disquisitiones Arithmeticae." Today, modular arithmetic is not just for solving number puzzles; it plays a crucial role in computer science, cryptography, and data transmission.