What is the Remainder When 3^35 is Divided by 5? GMAT Problem Solving

Question: What is the remainder when \(3^{35}\) is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4​

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
In order to solve this equation, candidates can opt for a cyclicity approach. Applying cyclicity can help candidates to yield an appropriate answer by undertaking a short-hand approach.
Below mentioned are the steps that will be followed for solving the mentioned problem:
The first step that candidates are to consider is what are the values when the power 35 is replaced by other numbers:

First; \(3^1\) yields the answer 3. Moreover, the remainder that is acquired after dividing 3 by 5 is 3.
Secondly; \(3^2\) gives us the answer 9. Thus, upon dividing 9 by 5 candidates can evidently see that the answer remainder is 4.
The third step that follows in this approach is; \(3^3\). Upon solving this, we get 27. Thus, by dividing 27 by 5, we get the remainder 2.
The fourth step that is followed in this approach is; \(3^4\). Upon solving this equation, the value that we get is 81. Thus, further proceeding with the division of 81 by 5, we get 1.
The fifth and final step that follows is \(3^5\). The answer that we get is 243 upon solving the mentioned equation. Further dividing the equation by 5, the remainder that gets is 3 again.

Thus, the cyclicity for this equation is 4.
Therefore upon dividing 35/4, we conclude the 3rd order of cyclicity of 3.

The remainder that we get is 2. 

Approach Solution 2:
The problem statement asks to find the remainder when \(3^{35}\) is divided by 5.

By dividing \(3^{35}\)  by 5 we get:
3³⁵/5 = (3³² * 3³)/5

         = [(3²)¹⁶ * 27]/5

         = (9¹⁶ * 27)/5

When 9 is divided by 5, we get: the remainder i.e R = -1
When 27 is divided by 5, we get the remainder i.e R = 2

Thus, overall remainder = (-1)¹⁶ * 2 = 2.

Hence, the remainder when \(3^{35}\) is divided by 5 = 2.

Approach Solution 3:
The problem statement asks to find the remainder when \(3^{35}\) is divided by 5.

We can solve the problem by finding the cycle of the unit's digit of power of 3 and then generalizing it.
Unit's digit of 3¹ = 3
Unit's digit of 3² = 9
Unit's digit of 3³ = 7
Unit's digit of 3⁴ = 1
Unit's digit of 3⁵ = 3

Therefore it can be analysed that the unit's digit of power of 3 repeats after every 4th number.
Hence, we need to divide 35 by 4 and check what is the remainder
=> 35 divided by 4 gives 3 remainder

=> 3³⁵ will have the same unit's digit as 3³= 7
=> Unit's digits of 3³⁵ = 7

But the remainder of 3³⁵ by 5 cannot be more than 5
=> Remainder = Remainder of 7 by 5 = 2

Hence, the remainder when \(3^{35}\) is divided by 5 = 2.

“What is the remainder when \(3^{35}\) is divided by 5 ”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide Quantitative Review 2022”. To solve the GMAT Problem Solving questions, the candidates must have a basic understanding of mathematics and calculations. The candidates can practise varieties of questions from the GMAT Quant practice papers that will help them to improve their mathematical knowledge.

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