What Is The Tens Digit Of 6^17? GMAT Problem Solving

Question: What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

“What is the tens digit of 6^17?''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This GMAT Quantitative question has been taken from the book “GMAT Official Advanced Questions”. GMAT quant section is devised to measure the scope of the candidates in using their quantitative knowledge. The candidates must have basic notions of maths that includes arithmetic, algebra and geometry. The candidates need to calculate the sum with accurate numerical verifications to solve GMAT Problem Solving questions. The mathematical problems based on the GMAT Quant topic in the problem-solving part can be solved with adequate quantitative skills.

Solution and Explanation:
Approach Solution 1:

The problem statement asks to find out the tens digit of 6^17.
Let’s derive the solution on the basis of pattern recognition.
The tens digit of 6 in integer power starting from 2 gets repeated in the pattern of 5: {3, 1, 9, 7, 5}, since 6^1 does not have any tens digit.
We get the pattern of 5 of the tens digit of 6 in integer in the following ways:
The tens digit of 6^2=36 is 3;
The tens digit of 6^3=216 is 1;
The tens digit of 6^4=...96 is 9 (this can be calculated by multiplying 16 by 6 to get …96 as the last two digits);
The tens digit of 6^5=...76 is 7 (this can be calculated by multiplying 96 by 6 to get ...76 as the last two digits);
The tens digit of 6^6=...56 is 5 (this can be calculated by multiplying 76 by 6 to get ...56 as the last two digits);
The tens digit of 6^7=...36 is 3 (this can be calculated by multiplying 56 by 6 to get ...36 as the last two digits). Thus we will get the same pattern again.
Therefore, we can say that 6^2, 6^7, 6^12, 6^17, 6^22, ... will hold the same tens digit of 3.

Correct Answer: (B)
Approach Solution 2:

The problem statement asks to find out the tens digit of 6^17.
We can deduce the problem by using a simpler and quicker way.
We can derive the equation 6^17 as (2×3)^17
In the next step, we can write, 2^17 × 3^17
Now, 2^17 can be split into 2^10 × 2^7 = 24 × 28 = 72
And 3^17 can be split into 3^16 × 3 = 81^4 ×3 = 21 × 3 = 63
Therefore, the last two digits = 72 x 63 = 36.
Thus, the tens digit of 6^17= 3

Correct Answer: (B)
Approach Solution 3:

The problem statement asks to find out the tens digit of 6^17.
We can solve the problem by the following method:
6^2= 36
Then, 6^3= 36∗ 6 = n16
Then, 6^5=6^2∗ 6^3 = 36∗ n16 = n76
It is required to just know the tens digits of the product of the numbers. We do not need to find the entire result of the product.
Therefore, from the above table, we can calculate the tens digit of 6^10 and 6^17.
Therefore, 6^10= 6^5 ∗ 6^5 = n76 ∗ n76 = n76
Therefore, 6^7= 6^5 ∗ 6^2 = n36 ∗ n76 = n36
Hence, 6^17 = 6^10 ∗ 6^7 = n76 ∗ n36 =n36
Thus, the tens digit of 6^17= 3

Correct Answer: (B)

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