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rainmaker: Senior | Next Rank: 100 Posts; Posts: 63; Joined: Mon Jul 20, 2009 11:48 am; GMAT Score:730

by rainmaker » Sat Aug 22, 2009 6:57 pm

Please explain your answer:

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

where ^ means "raised to the power of"

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Source: Beat The GMAT — Data Sufficiency | Sat Aug 22, 2009 6:57 pm

capnx: Master | Next Rank: 500 Posts; Posts: 268; Joined: Sat Aug 15, 2009 9:22 amFollowed by: 1 member; Thanked: 19 times; GMAT Score:700

by capnx » Sat Aug 22, 2009 7:40 pm

is it A? stmt 1 alone is sufficient.

since it's asking only the unit digit of n, then just looking at the unit digits:
243^x -->3^x
463^y -->3^y

so from the two lines: 3^x*3^y = 3^(x+y)
if x+y=7, we know thd unit digit of 3^7, so unit digit of n can be found

stmt 2, we know x, but don't know y, so not sufficient

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stilite: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Wed Aug 19, 2009 12:41 am

confusing solution

by stilite » Sun Aug 23, 2009 2:08 am

capnx wrote:is it A? stmt 1 alone is sufficient.

since it's asking only the unit digit of n, then just looking at the unit digits:
243^x -->3^x
463^y -->3^y

so from the two lines: 3^x*3^y = 3^(x+y)
if x+y=7, we know thd unit digit of 3^7, so unit digit of n can be found

stmt 2, we know x, but don't know y, so not sufficient

Capnx,

I'm just curious about your solution,...because it looks like you have simplified 243 down to 3, but 463 does not have "3" as a factor. So how did you finish the problem?

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capnx: Master | Next Rank: 500 Posts; Posts: 268; Joined: Sat Aug 15, 2009 9:22 amFollowed by: 1 member; Thanked: 19 times; GMAT Score:700

by capnx » Sun Aug 23, 2009 2:40 am

whether 3 is a factor is not needed. 3 is the last digit, so to get the last digit of the product of two numbers, all you need is the last digit.

ie: 13^x = ______
x = 1, = 13 --> 3^1=3
x = 2, = 169 --> 3^2=9
x = 3, = 2197 --> 3^3=27

so the power of the last digit only depends on the power of the last digit.

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stilite: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Wed Aug 19, 2009 12:41 am

by stilite » Sun Aug 23, 2009 3:09 am

Thanks for the clarification,... this, again, is the premiere problem I am having (as I posted on another forum question).

I am always missing the point of finding out whether the possibilities are "sufficient" versus knowing the actual answer. I need to burn this in my brain.

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chendawg: Master | Next Rank: 500 Posts; Posts: 163; Joined: Tue Nov 24, 2009 7:56 amFollowed by: 4 members; Location: Philadelphia; Thanked: 13 times; GMAT Score:660

by chendawg » Mon Feb 21, 2011 4:45 pm

I would go with A.

Capnx makes good points above with his explanation. Another way to do it would be to plug in values, for the non-algebraic way. We can see that all we need to do is find out how many times 3 is multiplied, so I made a list to find out when 3 repeats. Since the question only cares about the units digits, we can apply the last digit shortcut. 3^1 = 3, 3^2=9, 3^3=27, 3^4=81, 3^5= 243, 3^6=729 so we're sure we're back at the beginning, so we can just repeat as needed. Obviously don't multiply things out, but I did it to just illustrate the pattern.

Statement 1: x + y = 7

If x=1 and y=6, the last digit would be (3^1=3) * (3^6=729) = 3*9= 27 (using the last digit shortcut)

If x=2 and y=6, the last digit would be (3^2=9) * (3^5=243) = 9*3= 27

If x=3 and y=4, the last digit would be (3^3=7) * (3^4=81) = 7*1= 7

and so on....we can see that 7 will be the final digit for all of them. Sufficient.

Statement 2: x = 4 Doesn't tell us anything about y, thus insufficient.

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LalaB: Master | Next Rank: 500 Posts; Posts: 425; Joined: Wed Dec 08, 2010 9:00 amFollowed by: 7 members; Thanked: 56 times; GMAT Score:690

by LalaB » Mon Oct 31, 2011 12:45 am

IMHO A

we know, that

3^1=3(9 is a unit digit)
3^2=9(9 is a unit digit)
3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit)
3^5=243(3 is a unit digit)

so , now we r moving to stmnt 1

x+y=7

since we know that x and y are positive integers, then x colud be equal to 3;5;4;2 ,then y will be equal to 4;2;3;5 respectively.

so if x =3 ,and y =4 then the units digit of n will be 7 (since 3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit) 7*1 =7)

if x=5 then y=2 ,so the units digit of n will be 7 again (since 3^2=9(9 is a unit digit) and 3^5=243(3 is a unit digit) ; 9*3=27)

A is sufficient

B is not sufficient, since we have no info about y.

imho ,the trick of this question is the tendency of choosing C.

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LalaB: Master | Next Rank: 500 Posts; Posts: 425; Joined: Wed Dec 08, 2010 9:00 amFollowed by: 7 members; Thanked: 56 times; GMAT Score:690

by LalaB » Mon Oct 31, 2011 12:46 am

by the way, this question is time -consuming ( what is the source of it?

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pemdas: Legendary Member; Posts: 1084; Joined: Fri Apr 15, 2011 2:33 pmFollowed by: 21 members; Thanked: 158 times

by pemdas » Mon Oct 31, 2011 1:24 am

you are listing choices (variations of x+y=7) and then arbitrarily decide to choose x=3 and y=4, perhaps after grueling solution and understanding that all variations give the same answer.

Instead note please, the unit's digit of n will be resolved by finding x and y
st(1) suggests that x+y=7 which is only possible if x or y is at least 1 and/or x or y is at most 6 (x,y are positive integers per this q.). So the variations we have (3^1)*(3^6) or two more variations, like ((3^2)*(3^5), (3^3)*(3^4) listed by you can be quickly regrouped as 3^7, because we account for the last number=3 in both factors 243 and 463 and be it (3^1)*(3^6) or (3^3)*(3^4) this can be rewritten as 3^(x+y=7)

this way you save ton's of time and avoid lengthy solution for all variations, not mentioning arbitrary choice "let x=3 and y=4"

st(2) x=4 and y can be anything is Not Sufficient

rainmaker wrote:Please explain your answer:

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

where ^ means "raised to the power of"

LalaB wrote:IMHO A

we know, that

3^1=3(9 is a unit digit)
3^2=9(9 is a unit digit)
3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit)
3^5=243(3 is a unit digit)

so , now we r moving to stmnt 1

x+y=7

since we know that x and y are positive integers, then x colud be equal to 3;5;4;2 ,then y will be equal to 4;2;3;5 respectively.

so if x =3 ,and y =4 then the units digit of n will be 7 (since 3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit) 7*1 =7)

if x=5 then y=2 ,so the units digit of n will be 7 again (since 3^2=9(9 is a unit digit) and 3^5=243(3 is a unit digit) ; 9*3=27)

A is sufficient

B is not sufficient, since we have no info about y.

imho ,the trick of this question is the tendency of choosing C.

Success doesn't come overnight!

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LalaB: Master | Next Rank: 500 Posts; Posts: 425; Joined: Wed Dec 08, 2010 9:00 amFollowed by: 7 members; Thanked: 56 times; GMAT Score:690

by LalaB » Mon Oct 31, 2011 2:31 am

pemdas wrote: this can be rewritten as 3^(x+y=7)

pemdas,I got what u said .that was nice ) sometimes life is easier than my brain can imagine )

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HSPA: Legendary Member; Posts: 1101; Joined: Fri Jan 28, 2011 7:26 amFollowed by: 13 members; Thanked: 47 times; GMAT Score:640

by HSPA » Thu Nov 03, 2011 8:59 am

HI

Is 463 a prime number??

First take: 640 (50M, 27V) - RC needs 300% improvement
Second take: coming soon..
Regards,
HSPA.

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GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members; Location: Austin, TX; Thanked: 236 times; GMAT Score:770

by GmatMathPro » Thu Nov 03, 2011 10:58 am

HSPA wrote:HI

Is 463 a prime number??

Yup.

Pete Ackley
GMAT Math Pro
Free Online Tutoring Trial

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karthikpandian19: Legendary Member; Posts: 1665; Joined: Thu Nov 03, 2011 7:04 pmFollowed by: 70 members; Thanked: 165 times

by karthikpandian19 » Wed Jun 13, 2012 11:05 pm

pemdas,

Your solution is ofcourse the fastest one....thank you

pemdas wrote:you are listing choices (variations of x+y=7) and then arbitrarily decide to choose x=3 and y=4, perhaps after grueling solution and understanding that all variations give the same answer.

Instead note please, the unit's digit of n will be resolved by finding x and y
st(1) suggests that x+y=7 which is only possible if x or y is at least 1 and/or x or y is at most 6 (x,y are positive integers per this q.). So the variations we have (3^1)*(3^6) or two more variations, like ((3^2)*(3^5), (3^3)*(3^4) listed by you can be quickly regrouped as 3^7, because we account for the last number=3 in both factors 243 and 463 and be it (3^1)*(3^6) or (3^3)*(3^4) this can be rewritten as 3^(x+y=7)

this way you save ton's of time and avoid lengthy solution for all variations, not mentioning arbitrary choice "let x=3 and y=4"

st(2) x=4 and y can be anything is Not Sufficient

rainmaker wrote:Please explain your answer:

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

where ^ means "raised to the power of"

LalaB wrote:IMHO A

we know, that

3^1=3(9 is a unit digit)
3^2=9(9 is a unit digit)
3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit)
3^5=243(3 is a unit digit)

so , now we r moving to stmnt 1

x+y=7

since we know that x and y are positive integers, then x colud be equal to 3;5;4;2 ,then y will be equal to 4;2;3;5 respectively.

so if x =3 ,and y =4 then the units digit of n will be 7 (since 3^3=27 (7 is a unit digit)
3^4=81 (1 is a unit digit) 7*1 =7)

if x=5 then y=2 ,so the units digit of n will be 7 again (since 3^2=9(9 is a unit digit) and 3^5=243(3 is a unit digit) ; 9*3=27)

A is sufficient

B is not sufficient, since we have no info about y.

imho ,the trick of this question is the tendency of choosing C.

Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON

---If you find my post useful, click "Thank"

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rajeshsinghgmat: Master | Next Rank: 500 Posts; Posts: 171; Joined: Tue Jan 08, 2013 7:24 am; Thanked: 1 times

by rajeshsinghgmat » Wed May 01, 2013 2:44 am

D in answer.

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ndqv: Senior | Next Rank: 100 Posts; Posts: 45; Joined: Fri Aug 30, 2013 7:51 pm

by ndqv » Tue Dec 03, 2013 6:26 am

1) X+Y =7

Suppose X<Y
x=1, y=6 => units digit = 7
x=2, y=5 => units digit = 7
x=3, y=4 => units digit = 7

As 243 and 463 have the same units digits, the case x>y leads to the same result.

=> stat 1 is sufficient

2) Only know x => insufficient

*Another way: For stat 1): ignore 243 and 463. What matters here, the units digit of n, is exactly the same as the units digits of 243^x*243^y (=243^7) => units digit is 7

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