What is the Last Digit of \(3^{3^3}\)  GMAT Problem Solving

Question: What is the last digit of \(3^{3^3}\) ?

1. 1
2. 3
3. 6
4. 7
5. 9

“What is the last digit of \(3^{3^3}\) ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution 1:

There is only one solution approach to this problem.

If the exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:

\(a^{m^n}=a^{(m^n)}\) and not \((a^{m})^n\) , which on the other hand equals to \(a^{mn}\)

So, \((a^{m})^n=a^{mn};\)

\(a^{m^n}=a^{(m^n)}\) and not \((a^{m})^n\)

So according to the above given equation in the question, \(3^{3^3}=3^{(3^3)}=3^{(27)}\)

Next the unit’s digit of 3 in positive integer power repeats in pattern of 4: {3,9,7,1}.

So, the unit’s digit of \(3^{(27)}\) (27 = 4*6 + 3) is same as the unit’s digit of \(3^3\) , which is 7.

Correct Answer: D

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