What is the unit digit of 42^n?

When dealing with large powers of numbers, finding the unit digit can be simplified by focusing only on the unit digit of the base number. Here, the base number is 42. The unit digit of 42 is 2, so we can simplify our problem to finding the unit digit of 2 raised to the power of n.

Pattern of Unit Digits for Powers of 2

To find the unit digit of 2^n, let’s observe the pattern formed by the unit digits of the first few powers of 2:

• $2^1 = 2$ (unit digit is 2)
• $2^2 = 4$ (unit digit is 4)
• $2^3 = 8$ (unit digit is 8)
• $2^4 = 16$ (unit digit is 6)
• $2^5 = 32$ (unit digit is 2)
• $2^6 = 64$ (unit digit is 4)
• $2^7 = 128$ (unit digit is 8)
• $2^8 = 256$ (unit digit is 6)

We notice that the unit digits repeat every 4 powers: 2, 4, 8, 6. This cyclical pattern helps us determine the unit digit of any power of 2.

Finding the Unit Digit of 2^n

To find the unit digit of $2^n$, we need to determine the position of n in the cycle. This can be done using modulo arithmetic. Specifically, we look at $n bmod 4$:

• If $n bmod 4 = 1$, the unit digit is 2.
• If $n bmod 4 = 2$, the unit digit is 4.
• If $n bmod 4 = 3$, the unit digit is 8.
• If $n bmod 4 = 0$, the unit digit is 6.

Examples

1. For $n = 5$:
   • $5 bmod 4 = 1$, so the unit digit of $2^5$ is 2.
2. For $n = 10$:
   • $10 bmod 4 = 2$, so the unit digit of $2^{10}$ is 4.
3. For $n = 15$:
   • $15 bmod 4 = 3$, so the unit digit of $2^{15}$ is 8.
4. For $n = 20$:
   • $20 bmod 4 = 0$, so the unit digit of $2^{20}$ is 6.

Conclusion

By focusing on the unit digit of the base number and recognizing the cyclical pattern of the unit digits of powers of 2, we can easily determine the unit digit of $42^n$. The unit digit of $42^n$ is the same as the unit digit of $2^n$, which follows the repeating cycle: 2, 4, 8, 6. Identifying the position of n in this cycle allows us to find the unit digit quickly and accurately.

2. Brilliant – Cyclic Patterns

Citations

1. 1. Khan Academy – Modular Arithmetic
2. 3. Math is Fun – Powers of Numbers