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# If x = 54^n and y = 89^n and z =x*y . what is the units

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Senior Manager
Joined: 19 Feb 2004
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If x = 54^n and y = 89^n and z =x*y . what is the units [#permalink]  19 Feb 2004, 08:11
If x = 54^n and y = 89^n and z =x*y . what is the units digit of z?
a. n is a positive integer
b. n is an even integer
SVP
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z = (54 * 89 ) ^ n
The unit digit of this product is 6 and any number which has unit digit of 6 raised to any integer number will always be 6
However if N is not integer then the story is different.

Director
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anandnk wrote:
z = (54 * 89 ) ^ n
The unit digit of this product is 6 and any number which has unit digit of 6 raised to any integer number will always be 6
However if N is not integer then the story is different.

Anand! Are you sure !!
Director
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Rakesh : E
Anand : D

Only two answers are left; who want to grab the rest?
SVP
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Kudos [?]: 41 [0], given: 0

Sorry I realize my mistake.

Director
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anandnk wrote:
Sorry I realize my mistake.

Yes, I answer should be A
Senior Manager
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Perhaps correct [#permalink]  19 Feb 2004, 18:34
Hi, I shall give a try to, hopefully correct

Let us put that in to steps ..

Looking at the equations X=54^n ---- in this for every odd value of "n" the unit number will be 4 and for every even value of "n" the unit number will be 6. This pattern 4,6,4,6 wil follow ---

As for the second one y=89^n, a similar one, for all odd value of n the unit number will be 9 and for even it is 1, hence following the pattern 9,1,9,1,...

Important - The above statements hold good only when n is a positive integer and not negetaive

Now, let us look at the statements

Statement A - N is a positive integer ... emm not good enough to kow the unit of z as n can be odd or even.

Statement B - N is a even integer .. emmm not good enough to know the unit as n can be a negetive even interger or positive even integer.

Let us combine them and see A and B, if it is +ve and even then we will get the unit digit as 6 (6x1=6). There is no other possibilty. Hence the answer is C. I hope this is right.

Answer = C (both needed to solve)

Regards
Senior Manager
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good try, but you all missed on trying out 0 (is even) in statement 2.

Ans is A.
1. is sufficient . if n=1, x=54^n =54, y=89^n =89, and x*y will have a units digit of (4*9=36) 6.
If n = 2 , x= 54^2 = ...6, y= 89^2 = ...1, and x*y will have a unit's digit of (4*9=36) 6.

2. is insufficient . if n=2, x=54^2 = ...6, y =89^2 =.. and x*y will have a units digit of (9*4=36) 6.
However, if n= 0, x = 54 ^0 = 1, y =89^0 =1, and thus x*y will have a units digit of 1.
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