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# If a and b are positive integers and x = 4^a and y = 9^b, which of the

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Math Expert
Joined: 02 Sep 2009
Posts: 58327
If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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23 Oct 2014, 00:50
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Question Stats:

87% (01:24) correct 13% (01:43) wrong based on 494 sessions

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Tough and Tricky questions: Exponents.

If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

A. 1
B. 4
C. 5
D. 7
E. 8

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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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23 Oct 2014, 02:21
4
2
Bunuel wrote:

Tough and Tricky questions: Exponents.

If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

A. 1
B. 4
C. 5
D. 7
E. 8

unit digit of x=4^a will either be 4 or 6 (4 if a is odd and 6 if a is even)
and unit digit of y=9^b will either be 1 or 9 (9 if b is odd and 1 if b is even)
now we have total 4 possible combinations

1) a=odd, b=odd
unit digit=9*4=6
2)a=odd, b=even
unit digit=4*1=4
3) a=even,b=odd
unit digit=6*9=4
4)a=even, b=even
unit digit=6*1=6

thus, only two values are possible for the unit digits (4 or 6). out of these values only 4 is mentioned in the given choices. hence answer must be B.
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Math Expert
Joined: 02 Sep 2009
Posts: 58327
Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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23 Oct 2014, 00:53
Bunuel wrote:

Tough and Tricky questions: Exponents.

If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

A. 1
B. 4
C. 5
D. 7
E. 8

Check Units digits, exponents, remainders problems directory in our Special Questions Directory.
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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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23 Oct 2014, 23:02
2
Bunuel wrote:

Tough and Tricky questions: Exponents.

If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

A. 1
B. 4
C. 5
D. 7
E. 8

Only Units digit of X and Y contribute to the Units digit of XY.

Finding all values of "X" for positive values of "a"

When,
a=1; X=4
a=2; X=16
a=3; X=64........ Units digit repeat in a cycle of 4,6,4,6......

Similarly, Finding all values of "Y" for positive values of "b"

When,
b=1; Y=9
b=2; Y= 81
b=3; Y= 729....... Units digit repeat in cycle of 9,1,9,1....

So, possible value of Units digit of XY can be found by considering all possibilities of Units digit of X and Y

When X=4 and Y=9: Units digit of XY = 6
When X=4 and Y=1: Units digit of XY = 4
When X=6 and Y=9: Units digit of XY = 4
When X=6 and Y=1: Units digit of XY = 6

Only two possibilities for Units digit of XY i.e. 4 and 6.

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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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24 Oct 2014, 00:46
1
Bunuel wrote:

Tough and Tricky questions: Exponents.

If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

A. 1
B. 4
C. 5
D. 7
E. 8

Cyclicity of 4=2

4^1=4
4^2=16
4^3=64
Cycicity of 9 is 2
9^1=9
9^2=81
9^3=729

So possible options 4*9 =36 or 1*4=4

Ans B
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If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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26 Oct 2014, 19:32
$$4^1 = 4 .......................... 9^1 = 9$$

$$4^2 = 16 ......................... 9^2 = 81$$

$$4^3 = 64 ........................... 9^3 = 729$$

Total combinations possible are four which are (4*9) (4*1) (6*9) (6*1)

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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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05 Feb 2017, 14:00
x=4^a where a is a positive integer will have a units digit of 4 or 6

4^1 = 4 -> units digit 4
4^2 =16 -> units digit 6
4^3 = 64 -> units digit 4

y=9^b where b is a positive integer will have a units digit of 9 or 1.

9^1 = 9 -> units digit 9
9^2 = 81 -> units digit 1
9^3 =729 -> units digit 9

Product of xy will have units digit of 4*9=36=units digit 6 , 4*1=4=units digit 4 , 6*9=54=units digit 4 or 6*1=6=units digit 6.

So xy can have either units digit 4 or 6.

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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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12 Feb 2017, 15:26
a, b are positive integers
x = 4^a ; y = 9^b
a = 4 or 6
b = 1 or 9
xy = 4 or 6
B
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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the  [#permalink]

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Re: If a and b are positive integers and x = 4^a and y = 9^b, which of the   [#permalink] 17 May 2019, 04:33
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