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# If 243^x*463^y = n, where x and y are positive integers, wha

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

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05 Nov 2010, 04:05
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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

[Reveal] Spoiler:
The above question is from MGMAT. The answer given is A.

But i believe, even (X=4) will also give the solution.

As X=4, the power of 3 in the expression [(243)^x]X[(463)^y] would always be 3^4n and the units digit will always be 1.

Can some one comment on this. Bunuel your help would be very much appreciated.
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If 243^x*463^y = n, where x and y are positive integers, wha [#permalink]

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05 Nov 2010, 04:36
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mattapraveen wrote:
If [(243)^x]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

The above question is from MGMAT. The answer given is A.

But i believe, even (X=4) will also give the solution.

As X=4, the power of 3 in the expression [(243)^x]X[(463)^y] would always be 3^4n and the units digit will always be 1.

Can some one comment on this. Bunuel your help would be very much appreciated.

The units digit of $$243^x$$ equals to the units digit of $$3^x$$ and the units digit of $$463^y$$ equals to the units digit of $$3^y$$ (general rule). Hence the units digit of $$243^x*463^y$$ equals to the units digit of $$3^x*3^y=3^{x+y}$$. So knowing the value of $$x+y$$ is sufficient to determine the 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 the digit of $$3^3$$ which is $$7$$)

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

For example if $$y=1$$ then the units digit of $$3^{x+y}=3^5$$ will be 3, but if $$y=4$$ then the units digit of $$3^{x+y}=3^8$$ will be 1.

For more on this issue check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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Re: MGMAT DS Question [#permalink]

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05 Nov 2010, 04:44
mattapraveen wrote:
If [(243)^x]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

The above question is from MGMAT. The answer given is A.

But i believe, even (X=4) will also give the solution.

As X=4, the power of 3 in the expression [(243)^x]X[(463)^y] would always be 3^4n and the units digit will always be 1.

Can some one comment on this. Bunuel your help would be very much appreciated.

Let me try ... i am not even 1% what bunuel is ..

The reason why x=4 is insufficient because we dont know anything about Y, so if x=4 and y is 1, the unit digit will be 3, if y=2 then, unit digit will be 9 and so on.

So, i think it is necessary to have exact value of both the variants before reaching to the unit digit.

Had it been 5 as base , then it would not have made any difference because the unit digit is anyhow gonna be 5 only.

am open to comments .. correct me if i am wrong anywhere

HTH
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Status: Starting Work
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Joined: 20 Jun 2010
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Location: United Arab Emirates
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WE: Business Development (Energy and Utilities)
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Re: MGMAT DS Question [#permalink]

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05 Nov 2010, 04:59
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.

For example if y=1 then units digit of 3^{x+y}=3^5 will be 3, but if y=4 then units digit of 3^{x+y}=3^8 will be 1.

For more on this issue check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.

Thanks Bunuel. Yeah, got it. Some how got into an illusion that (X+Y)=(X*Y) . Hence was assuming when X=4, Y will be 4*Y.

Spotted my silly error.
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Re: If 243^x*463^y = n, where x and y are positive integers, wha [#permalink]

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31 Jul 2014, 09:39
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Re: If 243^x*463^y = n, where x and y are positive integers, wha   [#permalink] 31 Jul 2014, 09:39
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# If 243^x*463^y = n, where x and y are positive integers, wha

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