GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 03 Jul 2020, 05:45

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

Manager
Joined: 07 Feb 2010
Posts: 110
If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

02 Oct 2010, 03:44
4
36
00:00

Difficulty:

65% (hard)

Question Stats:

57% (01:36) correct 43% (01:45) wrong based on 799 sessions

### HideShow timer Statistics

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
Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

02 Oct 2010, 03:56
10
8
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.
_________________
##### General Discussion
Intern
Joined: 10 Jul 2010
Posts: 32
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Oct 2010, 08:50
Yup!! A it is....equation can be treated like 3^x*3^y hence (x+y)'s value can provide us the last digit...
Intern
Joined: 24 Feb 2014
Posts: 4
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Mar 2014, 04:18
I have a doubt. Cyclicity of unit digit of 3 is 4. Hence we know that every fourth power of 3 (3^4, 3^8, 3^12) will have the same unit digit, 1. Hence when option B says x = 4, knowing that x and y are positive integers, we know that xy will be a multiple of 4. Unit digit of 3^4k is always 1 isn't it? Shouldn't this be sufficient information?

Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Mar 2014, 04:21
1
siriusblack1106 wrote:
I have a doubt. Cyclicity of unit digit of 3 is 4. Hence we know that every fourth power of 3 (3^4, 3^8, 3^12) will have the same unit digit, 1. Hence when option B says x = 4, knowing that x and y are positive integers, we know that xy will be a multiple of 4. Unit digit of 3^4k is always 1 isn't it? Shouldn't this be sufficient information?

I think you are missing that $$3^x*3^y=3^{x+y}$$, so the exponent is x+y not xy.

Does this make sense?
_________________
Intern
Joined: 24 Feb 2014
Posts: 4
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Mar 2014, 04:26
Yes! Can't believe I just made that mistake. Such mistakes are gonna cost me. :/
Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Mar 2014, 04:28
siriusblack1106 wrote:
Yes! Can't believe I just made that mistake. Such mistakes are gonna cost me. :/

Yes, careless errors are the #1 cause of score drops on the GMAT! They cause you to miss easier questions, hurting your score a lot more than not know how to solve the harder ones. So, be more careful, before you submit your answer, double-check that it’s the answer to the proper question.
_________________
Intern
Joined: 18 May 2014
Posts: 49
Location: United States
Concentration: General Management, Other
GMAT Date: 07-31-2014
GPA: 3.99
WE: Analyst (Consulting)
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

18 May 2014, 08:53
1
243 = 3^5

463 ends with a 3. So we have to know how many times we will multiply 3's at the end of each numbers.

1) 7 times - SUF
2) we dont know Y - INSUF

Choose (a)
Manager
Joined: 15 Aug 2013
Posts: 223
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

02 Nov 2014, 17:48
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

Hi Bunuel,

But surely shouldn't it matter if 3 is raised to 1 and or 6? Meaning, if it's 3^3 + 3^4 = 7 + 1 = 8. But, if its 3^2+3^5 = 9 + 3 = 12, units of 2. Doesn't that yield insufficient?
Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

03 Nov 2014, 00:50
russ9 wrote:
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

Hi Bunuel,

But surely shouldn't it matter if 3 is raised to 1 and or 6? Meaning, if it's 3^3 + 3^4 = 7 + 1 = 8. But, if its 3^2+3^5 = 9 + 3 = 12, units of 2. Doesn't that yield insufficient?

Are you sure you are reading the question correctly? It's 243^x*463^y, 243^x multiplied by 463^y not 243^x + 463^y...
_________________
Intern
Joined: 04 Nov 2015
Posts: 44
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

28 Jul 2016, 14:01
Well I don't agree the answer should be indeed D.
Option 1 suggests x+y=7 this can have multiple x and y combinations like (1,6) (2,5) (4,3) and so on so the units digit of 243^x and 463^y will differ .
Intern
Joined: 17 Mar 2016
Posts: 13
Location: Singapore
GPA: 3.5
WE: Business Development (Energy and Utilities)
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

28 Jul 2016, 17:37
sandeep211986 wrote:
Well I don't agree the answer should be indeed D.
Option 1 suggests x+y=7 this can have multiple x and y combinations like (1,6) (2,5) (4,3) and so on so the units digit of 243^x and 463^y will differ .

Hey Buddy,

All the combinations, for x+y=7, will yield the same units digit. Consider the following
x=1,y=6
243*463*463*463*463*463*463 ~~ To find units digit we just need 3^1 * 3^6 = 3^7, i.e 7 (units digit of 2187)

Same goes with other combinations. 3^7 ends up deciding the units digit.

Should the question would have been something like, 245^x * 463^y = n, the combinations of different values of x & y would have yielded different units digits.

Hope that clears
Senior Manager
Joined: 13 Oct 2016
Posts: 352
GPA: 3.98
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

20 Dec 2016, 14:25
amandeep_k 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

I can't get how the answer can be A.

Had it been $$243^x * 463^y$$. In this case we'll get same base 3 and we can add the powers

$$3^0*3^7$$
$$3^1*3^6$$
$$3^2*3^5$$
...

In each case we'll get $$3^7$$ which units digit we can identify. That that will be sufficient.

But we have 3*x*3*y = 9*x*y which can take any values. Not sufficient. I can't get the idea how answer can be A.

Please correct me if I'm wrong.
Intern
Status: Fighting Again to bell the CAT
Joined: 28 Aug 2016
Posts: 33
Location: India
GMAT 1: 640 Q49 V30
GMAT 2: 710 Q50 V35
GPA: 3.61
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

20 Dec 2016, 14:39
vitaliyGMAT wrote:
amandeep_k 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

I can't get how the answer can be A.

Had it been $$243^x * 463^y$$. In this case we'll get same base 3 and we can add the powers

$$3^0*3^7$$
$$3^1*3^6$$
$$3^2*3^5$$
...

In each case we'll get $$3^7$$ which units digit we can identify. That that will be sufficient.

But we have 3*x*3*y = 9*x*y which can take any values. Not sufficient. I can't get the idea how answer can be A.

Please correct me if I'm wrong.

Hi, You are right. my question was wrong. Sorry for the inconvenience.
Senior Manager
Joined: 15 Jan 2017
Posts: 314
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

18 Dec 2017, 01:58
1
A.
If we map the cyclicity of 3--> with x + y values --> 3 and 4/ 4 and 3/ 1 and 6; 6 and 1/ 5 and 2; 2 and 5 [3,9,7,1,3,9,7,1...] --> the answer is always 7. [7*1 = 7; 3*9 = _7..so on]

St 2 . No value for y
Manager
Joined: 01 Apr 2020
Posts: 80
Location: India
If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

13 May 2020, 19:40
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

I don't think you can do this $$3^x*3^y=3^{x+y}$$
Because of the bases i.e. 243 ≠ 463

Can you explain please why did you do this?
Thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

13 May 2020, 23:01
D4kshGargas wrote:
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

I don't think you can do this $$3^x*3^y=3^{x+y}$$
Because of the bases i.e. 243 ≠ 463

Can you explain please why did you do this?
Thanks!

I think I explained this in the highlighted part. No?
_________________
Manager
Joined: 01 Apr 2020
Posts: 80
Location: India
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

14 May 2020, 19:27
Bunuel wrote:
D4kshGargas wrote:
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

I don't think you can do this $$3^x*3^y=3^{x+y}$$
Because of the bases i.e. 243 ≠ 463

Can you explain please why did you do this?
Thanks!

I think I explained this in the highlighted part. No?

Yeah, but I couldn't intuitively understand it back then. Thanks tho
Manager
Joined: 22 Jan 2020
Posts: 73
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

01 Jun 2020, 08:55
Bunuel wrote:
D4kshGargas wrote:
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

I don't think you can do this $$3^x*3^y=3^{x+y}$$
Because of the bases i.e. 243 ≠ 463

Can you explain please why did you do this?
Thanks!

I think I explained this in the highlighted part. No?

I'm not sure this is right

x+y = 7 will give you 3 different combinations of x and y to add up to 7:

1&6
2&5
3&4

With cyclicality of 4 you can get the same answer for the first two pairs, but you don't get the same answer for the third pair.

cyclicity for 3 is: 3971

remainder for 1 & 6 are follows : 3 + 9
remainder for 2 & 5 are follows : 9 + 3

but remainder for the third is different:

remainder for 3 & 4 are follows : 7 + 1
Math Expert
Joined: 02 Sep 2009
Posts: 64938
Re: If 243^x*463^y = n, where x and y are positive integers  [#permalink]

### Show Tags

01 Jun 2020, 09:12
Andrewcoleman wrote:
Bunuel wrote:
D4kshGargas wrote:
Bunuel wrote:
If $$243^x*463^y =n$$, where x and y are positive integers, what is the units digit of n?

The units digit of $$243^x$$ is the same as the units digit of $$3^x$$ and similarly the units digit of $$463^y$$ is the same as the units digit of $$3^y$$, so 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 units digit of $$3$$ in integer power 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.

Hope it helps.

I don't think you can do this $$3^x*3^y=3^{x+y}$$
Because of the bases i.e. 243 ≠ 463

Can you explain please why did you do this?
Thanks!

I think I explained this in the highlighted part. No?

I'm not sure this is right

x+y = 7 will give you 3 different combinations of x and y to add up to 7:

1&6
2&5
3&4

With cyclicality of 4 you can get the same answer for the first two pairs, but you don't get the same answer for the third pair.

cyclicity for 3 is: 3971

remainder for 1 & 6 are follows : 3 + 9
remainder for 2 & 5 are follows : 9 + 3

but remainder for the third is different:

remainder for 3 & 4 are follows : 7 + 1

I'm not sure I understand what you mean. Are you saying that the units digit of 3^(3+4) is not the same as the units digit of 3^3*3^4? If so, then since 3^(3+4) = 3^3*3^4, the untis digit are the same (both equal to 2187).
_________________
Re: If 243^x*463^y = n, where x and y are positive integers   [#permalink] 01 Jun 2020, 09:12