What is the Units Digit of the Solution to \(177^{28} −133^{23}\) GMAT Problem Solving

Question: What is the units digit of the solution to \(177^{28} −133^{23}\)?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

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

The problem statement asks to find the units digit of the solution: 177^28 - 133^23

To solve the question, we need to diminish the entire number so that we could calculate it easily.
Therefore, we can say,
177^28 - 133^23 can be written as 7^28 - 3^23 which will have the same units digit.

Therefore, both the numbers 7 and 3 have a cyclicity of 4, that is, their powers repeat the units digit after every 4th power
Hence, 7^28 has the same units digit as 7^4, which is 1
In the same way, 3^23 has the same units digit as 3^3, which is 7

Therefore, we get
xx...xx1 - xx....xx7 = xx...xx4
Hence, as the Unit digit of the first term is less
Therefore, the unit digit of the result will be => 11-7 => 4
Thus, the unit digit of the solution to 177^28 - 133^23 is 4.

Approach Solution 2:

The problem statement asks to find the units digit of the solution: 177^28 - 133^23.
We can derive the solution as:
7^1 = units digit 7
7^2 = units digit 9
7^3 = units digit 3
7^4 = units digit 1
7^5= units digit is 7.....incremental powers after 4th power of 7, the units digit is repeated.
Therefore, we can say 177^28 => Units digit = (7^4)^7=7^28. Therefore the number will have unit digit 1

Similarly, we get:
3^1 = units digit 3
3^2 = units digit 9
3^3 = units digit 7
3^4 = units digit 1
3^5= Units is 3.....incremental powers after 4th power of 3, the units digit is repeated.

Therefore, we can say, 133^23 = 133^20+3.

The unit digit of 133^20 is 1 but we need ^23. Therefore, multiplying 3 times implies that the units digit of 3^3 which is 7
Therefore, 133^23 will have the unit digit as 7

Therefore to find the solution we need to merge the two and take the difference of units digit.
1 - 7. To do this, we need to borrow one from whatever tens place which equals 11-7 = 4.
Thus, the unit digit of the solution to 177^28 - 133^23 is 4.

Approach Solution 3:

The problem statement asks to find the units digit of the solution: 177^28 - 133^23.

Since we need to find the unit digit, we should ignore tens and higher digits.
177^28 - 133^23 = Unit digit of 7^28 - unit digit of 3^23
= Unit digit of (7^4 )^7 - unit digit of {(3^4)^5 x (3^3)}

Since the unit digit of 7^4 = unit digit of 49^2 = unit digit of 9^2 =1
= Unit digit of (1^4 )^7 - unit digit of {(81)^5 x 27}
= 1 - unit digit of (1^5 x 27)
= 1 - 7 = 11-7 =4
Since unit digit of (1-7) = unit digit of (11 - 7)
Thus, the unit digit of the solution to 177^28 - 133^23 is 4.

“What is the units digit of the solution to 177^28 −133^ 23?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “501 GMAT Questions”. GMAT Problem Solving questions help the candidates to go through every piece of information in order to crack numerical problems. The candidates can practice more questions from the GMAT Quant practice papers to enhance their mathematical knowledge and learning.

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