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# M07-20

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Math Expert
Joined: 02 Sep 2009
Posts: 55638

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16 Sep 2014, 00:35
00:00

Difficulty:

35% (medium)

Question Stats:

68% (00:58) correct 32% (00:59) wrong based on 53 sessions

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

A. 1
B. 3
C. 4
D. 6
E. 8

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 55638

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16 Sep 2014, 00:35
Official Solution:

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

A. 1
B. 3
C. 4
D. 6
E. 8

First of all $$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$, 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 $$3*6^3-2$$ is $$8-2=6$$.

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Intern
Joined: 18 Jan 2014
Posts: 11
GMAT 1: 640 Q49 V28
GPA: 3.5
WE: Operations (Energy and Utilities)

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23 Dec 2014, 10:05
I think this question is poor and not helpful.
It should be mentioned here that @ is considered as function and not as a digit. Or instead of "http://gmatclub.com/forum/memberlist.php?mode=viewprofile&un= expression f(x) could have been used. Please consider.
Board of Directors
Joined: 17 Jul 2014
Posts: 2547
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)

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31 Oct 2015, 21:13
$$4^4$$ $$/$$ $$2*4^2 - 2$$
$$2^8/2^5 -2$$
$$2^3 = 8 -2 = 6$$
$$6^6/2*6^2$$
$$2^6*3^6 / 2^3 * 3^2 -2$$
$$2^3*3^4 -2$$
$$2^3 = 8$$
$$3^4 = 81$$
81*8 = 648 -2 = 646[/m] so units digit 6.
Intern
Joined: 26 May 2014
Posts: 39
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

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10 Aug 2016, 13:57
We find out that @4 = 6

now while solving @6 : (6^4)/2 - 2

6^1= units digit 6
6^2 = units digit 6
..
..
6^4= units digit 6

Now, when units digit 6 is divided by 2 we get units digit 3.

Units digit 3 -2=1

What did i do wrong?
Math Expert
Joined: 02 Sep 2009
Posts: 55638

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11 Aug 2016, 02:05
devbond wrote:
We find out that @4 = 6

now while solving @6 : (6^4)/2 - 2

6^1= units digit 6
6^2 = units digit 6
..
..
6^4= units digit 6

Now, when units digit 6 is divided by 2 we get units digit 3.

Units digit 3 -2=1

What did i do wrong?

6/2 = 3, so yes the units digit is 3 but 6^2/2 = 18, the units digit is 8.
_________________
Intern
Joined: 09 Apr 2018
Posts: 25
Location: India
Schools: IIMA PGPX"20
GPA: 3.5

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23 Dec 2018, 07:23
Bunuel wrote:
Official Solution:

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

A. 1
B. 3
C. 4
D. 6
E. 8

First of all $$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$, 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 $$3*6^3-2$$ is $$8-2=6$$.

.

how u got this:
$$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$
Math Expert
Joined: 02 Sep 2009
Posts: 55638

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23 Dec 2018, 12:01
SUNILAA wrote:
Bunuel wrote:
Official Solution:

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

A. 1
B. 3
C. 4
D. 6
E. 8

First of all $$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$, 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 $$3*6^3-2$$ is $$8-2=6$$.

.

how u got this:
$$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$

$$\frac{a^y}{a^y}=a^{x-y}$$, so $$\frac{x^x}{x^2}=x^{x-2}$$
_________________
Intern
Joined: 09 Apr 2018
Posts: 25
Location: India
Schools: IIMA PGPX"20
GPA: 3.5

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27 Dec 2018, 08:45
Bunuel wrote:
SUNILAA wrote:
Bunuel wrote:
Official Solution:

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

A. 1
B. 3
C. 4
D. 6
E. 8

First of all $$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$, 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 $$3*6^3-2$$ is $$8-2=6$$.

.

how u got this:
$$\frac{x^x}{2x^2}-2 = \frac{x^{x-2}}{2}-2$$

$$\frac{a^y}{a^y}=a^{x-y}$$, so $$\frac{x^x}{x^2}=x^{x-2}$$

thanks
Re: M07-20   [#permalink] 27 Dec 2018, 08:45
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# M07-20

Moderators: chetan2u, Bunuel