If @\(x=\frac{x^2}{2x^2}-2\), What is the Units Digit of @(@4)? GMAT Problem Solving

Question: If @ \(x=\frac{x^2}{2x^2}-2\) , what is the units digit of @ (@4)?

1. 1
2. 3
3. 4
4. 6
5. 8

“If @ \(x=\frac{x^2}{2x^2}-2\) , what is the units digit of @ (@4)?”- 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.

Answer

Approach Solution 1

Firstly, @ \(\frac{x^x}{2x^2}-2=\frac{x^{x-2}}2-2\)

So, @4 = \(\frac{4^{4-2}}{2}-2=6\) ;

Next, @6 = \(\frac{6^{6-2}}{2}-2=\frac{6^4}{2}-2=\frac{6*6^3}{2}-2=3*6^3-2\)

Now, the units digit of \(6^3\) is 6

Thus, the units digit of \(3*6^3\) is 8, (3 * 6 = 18)

So, the units digit of \(3*6^3-2\) = 6

Correct option: D

Approach Solution 2

We have,

\(\frac{4^4}{2*4^2}-2\)

\(\frac{2^8}{2^5}-2\)

\(2^3=8-2=6\)

\(\frac{6^6}{2}*6^2=\frac{2^6*3^6}{2^3*3^2}-2\)

\(2^3*3^4-2\)

\(2^3=8\)

\(3^4=81\)

81 * 8 = 648 – 2 = 646 [/m]

So unit digit = 6

Correct option: D

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