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If (243)^x(463)^y = n, where x and y are positive integers,

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If (243)^x(463)^y = n, where x and y are positive integers, [#permalink]

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07 Apr 2010, 23:37
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If (243)^x(463)^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7
(2) x = 4

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-243-x-463-y-n-where-x-and-y-are-positive-integers-102054.html
[Reveal] Spoiler: OA
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07 Apr 2010, 23:43
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).
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08 Apr 2010, 00:23
nickk wrote:
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).

So how did you find the values of x & y from Stmt 1??
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08 Apr 2010, 00:59
Hussain15 wrote:
nickk wrote:
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).

So how did you find the values of x & y from Stmt 1??

Well we don't know the values of X and Y individually, but all we need to know is how many times a number with 3 in the units digit is multiplied by itself. Since X and Y are both exponents of such numbers, knowing x+y is sufficient.

Of course I might be wrong so the OA would be appreciated.
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08 Apr 2010, 02:02
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KUDOS
In this case, since the base number is different i.e 243 & 463, it makes sense to use the various combinations of x & y:
1. 1&6 or 6&1
2. 2&5 or 5&2
3. 3&4 or 4&3

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Substituting n in the above combinations and multiplying the ending unit digits of each of these numbers, we get the same unit digit i.e., 7.

Choice (B), keeps the n open for y, so the unit digit of the resultant number may vary.

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08 Apr 2010, 02:52
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Expert's post
rohitgoel15 wrote:
If (243)^x(463)^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7
(2) x = 4

Units digit of $$243^x$$ equals to units digit of $$3^x$$ and units digit of $$463^y$$ equals to units digit of $$3^y$$ (general rule). Hence units digit of $$243^x*463^y$$ equals to units digit of $$3^x*3^y=3^{x+y}$$. So knowing the value of $$x+y$$ is sufficient to determine units digit of $$n$$.

(1) $$x+y=7$$. Sufficient. (As cyclicity of $$3$$ is $$4$$, units digit of $$3^7$$ would be the same as of units digit of $$3^3$$ which is $$7$$)

(2) $$x=4$$. No info about $$y$$. Not sufficient.

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08 Apr 2010, 02:54
Bunuel can you please take a look at probability-question-84062.html
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08 Apr 2010, 03:57
Kudos
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09 Apr 2010, 23:22
great question ....
I made the mistake of choosing C.
Re: Units digit   [#permalink] 09 Apr 2010, 23:22
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