What is the Units Digit of 3^66? GMAT Problem Solving

Question: What is the units digit of 3^66?

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

Correct Answer: B
Solution and Explanation:
Approach Solution 1:

You must use the information supplied in the question to solve this GMAT problem. These concerns cover numerous mathematical disciplines. This question concerns probability. Because of how the options are presented, it is difficult to select the optimal option. Candidates must comprehend the correct approach to obtaining the required response. Only one of the five supplied alternatives is correct.

The digits' cyclicity is listed below.
0 has a cyclicity of 1, 1, 5, and 6.
The perfect squares 4 and 9 have a cyclicity of two.
The cyclicity of all other numbers, including 2, 3, 7, and 8, is 4.

To determine the unit position of a^b, where a denotes the base and b the power.
Divide the power by the base's cyclicity to determine the remaining amount.
Here, two cases come up.

Case 1: If there is no leftover.
The final digit for bases with the prefixes 2, 4, or 8 is 6.
The final digit for bases with the prefixes 3, 7, or 9 is 1.

Example 2: The last digit will be the base to the power of the remainder if the remainder is not equal to 0.
Find the unit's place of 3^66 in the stated question.
The power is 66 and the base is 3, with cyclicity of 4.
R66/2 = 2
The answer is given by raising the base to the power, which is = 3^2 = 9.

E is the correct answer.

Approach Solution 2:

You must use the information supplied in the question to solve this GMAT problem. These concerns cover numerous mathematical disciplines. This question concerns probability. Because of how the options are presented, it is difficult to select the optimal option. Candidates must comprehend the correct approach to obtaining the required response. Only one of the five supplied alternatives is correct.

We can write 3^66 = 3^64 * 3^2
1 will be the unit digit for 3^64.
and the units digit for 32 will be 9
1*9 = 9

The solution is E.

Approach Solution 3:

You must use the information supplied in the question to solve this GMAT problem. These concerns cover numerous mathematical disciplines. This question concerns probability. Because of how the options are presented, it is difficult to select the optimal option. Candidates must comprehend the correct approach for obtaining the required response. Only one of the five supplied alternatives is correct.

Search for a pattern

3^1 = 3
3^2 = 9
3^3 = 7
3^4 = 1
3^5 = 3 (Repeats here)

1+4n:3
n=16, then 1+4(16) Equals 65.
So 3^65 = 3.
And 3^66 = 9.

Hence E is the correct answer.

“What is the units digit of 3^66?" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”. To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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