Can You Use Patterns to Find the Last Two Digits of Powers of 4?

[sparsh]

How does one solve problems like these :

finding last two digits of 4^ 300

 

[Tide]

One simple way is to look for patterns. For example, the last digit of powers of 4 alternate between 4 and 6. Can you relate that to the number of factors of 4?

 

[Architeuthis Dux]

Yes, patterns can be used to find the last two digits of powers of 4. One way to solve a problem like finding the last two digits of 4^300 is to look for a pattern in the last two digits of powers of 4.

For example, the last two digits of 4^1 is 04, 4^2 is 16, 4^3 is 64, and 4^4 is 56. Notice that the last two digits repeat in a pattern of 04, 16, 64, 56. This pattern continues for every power of 4, with the last two digits alternating between 04 and 56.

To find the last two digits of 4^300, we can divide 300 by 4 to get a remainder of 0. This means that the last two digits will be the same as the last two digits of 4^4, which is 56. Therefore, the last two digits of 4^300 is 56.

Another way to solve this problem is to use the concept of cyclicity. The last two digits of powers of 4 have a cyclicity of 20, meaning that the last two digits will repeat every 20 powers. So to find the last two digits of 4^300, we can divide 300 by 20 to get a remainder of 0. This means that the last two digits will be the same as the last two digits of 4^20, which is 56.

In conclusion, patterns and cyclicity can be used to find the last two digits of powers of 4. By recognizing patterns and using the concept of cyclicity, we can easily solve problems like finding the last two digits of 4^300. This approach can also be applied to finding the last two digits of powers of other numbers.