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units digit

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units digit

by ddm » Tue Aug 05, 2008 3:49 pm
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?

1
4
5
7
8
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by malolakrupa » Tue Aug 05, 2008 4:30 pm
x= 4^a = (2^2)^a= (2^a)^2
similarly y = 9^b = (3^b)^2

since a^2 * b^2 = (a*b)^2

Hence xy can be thought of as (2^a*3^b)^2

Additionally if you look at the basic squares ,

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 47
8^2 = 64
9^2 = 81
10^2 = 100


you can have 1 in the units digit when 2^a*3^b either equates to 1 or 9 . So A can be eliminated.

B is possible when a = 1 and b = 2 or vice versa.

C can be eliminated due to the same reason as A being eliminated.

you can eliminate D and E as non of the basic squares have 7 or 8 in their units digit.

Hence B is the ANSWER
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by parallel_chase » Tue Aug 05, 2008 4:34 pm
unit digit of 4^a can be either 6 or 4
unit digit of 9^b can be either 9 or 1

Therefore unit digit of xy can either be 4 or 6

Hence B is the answer.
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by malolakrupa » Tue Aug 05, 2008 4:36 pm
Need to clarify D is eliminated because , 2^a*3^b will never equate to a number having 7 in its units digit.
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