Determining the Units Digit of k when k 3^3^3^…^3 with 100 Tetrations

In this article, we will delve into the problem of determining the units digit of a number ( k ), where ( k 3^{3^{3^{ldots^{3}}}} ) (with 100 threes in the exponentiation). We will explore the mathematics behind this intricate problem, using concepts like tetration, modulus arithmetic, and Fermat's Little Theorem.

Understanding Tetration and the Problem

This problem deals with tetration, which is the operation of iterated exponentiation. In simpler terms, it is repeated exponentiation, where each exponent is itself an exponentiation. For instance, ( 3^{3^{3}} ) (with two threes) is a base of 3 raised to the power of ( 3^3 ).

Breaking Down the Problem with Modular Arithmetic

The key to solving this problem is to understand the behavior of the units digit of powers of 3. This involves a deep dive into modular arithmetic and the use of properties of numbers, particularly Fermat's Little Theorem.

Modular Arithmetic Basics

Modular arithmetic focuses on the remainders of division. For example, ( 3^4 equiv 1 mod 10 ). This means that any fourth power of 3, when divided by 10, leaves a remainder of 1. This is a crucial observation for solving our problem.

Tetration and Commuting Modulo 4

First, let's consider the exponent itself, which is also a power of 3. We need to find ( r mod 4 ), where ( r 3^{s} ). Since any power of 3 is odd, and ( s ) is itself an odd power of 3, we can say:

( s equiv 3 mod 4 ) ( r 3^s equiv 3^3 equiv -1 equiv 3 mod 4 )

This means that the exponent ( r ) is congruent to 3 modulo 4, which is significant for the next step.

Applying Fermat's Little Theorem

According to Fermat's Little Theorem, ( a^{p-1} equiv 1 mod p ) for any integer ( a ) and a prime number ( p ). Here, ( p 5 ), and thus:

( 3^4 equiv 1 mod 10 )

Given ( r equiv 3 mod 4 ), we can express ( r ) as ( 4n 3 ) for some integer ( n ). Therefore:

( 3^r 3^{4n 3} (3^4)^n cdot 3^3 equiv 1^n cdot 3^3 equiv 3^3 mod 10 )

Since ( 3^3 27 ), and the units digit of 27 is 7, we conclude:

( 3^r equiv 7 mod 10 )

Hence, the units digit of ( k ) is 7.

Pattern Recognition and Proof

To confirm this result, let's observe the pattern in the units digits of the powers of 3:

( 3^1 3 ) (units digit: 3) ( 3^2 9 ) (units digit: 9) ( 3^3 27 ) (units digit: 7) ( 3^4 81 ) (units digit: 1) ( 3^5 243 ) (units digit: 3) ( 3^6 729 ) (units digit: 9) ( 3^7 2187 ) (units digit: 7) ( 3^8 6561 ) (units digit: 1)

As we can see, the units digits repeat every 4 powers of 3. Since the exponent ( s ) is an odd number, the units digit of ( 3^s ) will be the same as the units digit of ( 3^3 ), which is 7.

Conclusion and Final Answer

Therefore, the units digit of ( k ), where ( k 3^{3^{3^{ldots^{3}}}} ) with 100 threes, is 7. This conclusion is supported by both the modular arithmetic approach using Fermat's Little Theorem and the observed pattern in the units digits of powers of 3.

Key Takeaways

Tetration involves iterated exponentiation. Modular arithmetic helps in finding patterns in the units digits. Fermat's Little Theorem simplifies calculations in modular arithmetic.

By understanding these concepts and their applications, you can solve similar complex problems involving tetration and units digits.