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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] New post 05 Nov 2010, 03:05
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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.
[Reveal] Spoiler: OA
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If 243^x*463^y = n, where x and y are positive integers, wha [#permalink] New post 05 Nov 2010, 03:36
Expert's post
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.

Answer: A.

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] New post 05 Nov 2010, 03:44
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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Re: MGMAT DS Question [#permalink] New post 05 Nov 2010, 03:59
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.

Answer: A.

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] New post 31 Jul 2014, 08:39
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Re: If 243^x*463^y = n, where x and y are positive integers, wha   [#permalink] 31 Jul 2014, 08:39
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