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Here is my take on the Problem :

If 4^w = n, what is the units digit of n?

1. w is an even integer
2. w > 0

Before we get into the two statements, we need to understand what we have :

We know that Unit digit of 4^w will follow a pattern (for all w>0 )
And for W=0 it becomes 1 and for negative numbers we cannot predict anything.

When w>0 ; we have a pattern for unit's digits:
4^1 = 4
4^2 = 6
4^3 = 4
4^4 = 6
.
.........

a) If "w" is odd -> the unit digit is 4
b) if "w" is even -> The unit digit is 6.


Now lets jump into the statements given :

(1) W is a even integers We cannot be very sure because "w" can also be "0" or negative numbers.

INSUFFICIENT

(2) w>0
This alone again will not help us to determine anything about the unit's digit.

INSUFFICIENT


(1) + (2) Combining both the statements

We come to know "W" cannot be zero neither negative and "W" is even.

Hence, we can conclude the unit's digit for 4^w is 6.

Both the statements together is required to answer this question.Hence the Correct answer choice is C

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Ans is D IMHO

Solution:
4^0= units place is 1
4^1= units place is 4
4^2= units place is 6
4^3=units place is 4
4^4=units place is 6 and so on.

We can observre a trend here.
Statement 1 : W is even means 0 or 2 or 4 e.t.c In this case units place can be either 1 or 6

St 2 : w > 0
So units place can be 4 or 6.

Both statement alone give us an answer.
Both combined also give us a single answer as 6.

Hence the above answer based on given options.

Posted from GMAT ToolKit

The answer is (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

1) w is an even integer. If we pick w to be negative then the units digit is 0 and if w is positive the answer is 6 so NOT sufficient
2) w > 0. If we pick w to be 1 we get unit as 4, w = 2 we get unit as 6. so it is NOT sufficient

If we combine them to have w is even and positive then the only possible answer is 6. so both options combined are SUFFICIENT. therefore the answer is The answer is (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

My answer would be C.

As per statement 1:
w is even number. So it can be 0, 2, 4,....
For all even number except 0, unit digit of n is 6.
For 0 unit digit for n is 1. so not sufficient.

As per statement 2:
w is any number greater than 0. so unit digit of n changes with different value of w.
for example:
when w is 1 then unit digit of n is 4,
when w is 2 then unit digit of n is 6,
when w is 3 then unit digit of n is 4,
so not sufficient.

When statement 1 and 2 are evaluated together then w can be 2,4,6,8.... For all these values of w, unit digit of n is 6.

Thus option C is correct.

The answer is A....

statement one is sufficient as it says that w is an even integer ..So for example w is 2 then = 16 so the unit digit is 6

Next example w is 4 = 96 again unit digit is 6

even if you solve w to be 6 the unit digit will be 6 everytime...

statement 2 is not sufficient because if w>0 w can be 1 in that case unit digit is 4

w can be 2 = 16 unit digit will be 6...So not sufficient..

Hence answer is A

Ans according to me is C

Soln: 4^w=n

Let us simplify this statement a bit-
plugging different values for w:

4^w=Units Digit(w is a positive integer)
4^1=4
4^2=6
4^3=4
4^4=6
4^5=4
4^6=6
Clearly, units digit is 4 if w is odd and units digit is 6 if w is even

Now consider
4^w=Units digit(w is a negative integer)
4^-1=0.25,units digit=0
4^-2=0.0625,units digit=0
4^-3=0.015625,units digit=0
Clearly here units digit is 0

Statement I= w is even integer:Insufficient since we dont know if w is positive or negative
Statement II=w>0:Insufficient since we don't know if w is odd or even

I+II=W is a positive ,even integer
Hence Units digit is 6
Therefore C

ANSWER: C

4^w=n..... if no other data is given, we can determine the fact that the units digit is 4 or 6 **. OR if w=0, n=1

A negative 'w' will give fractional values of n ; whose 'unit digit' doesnt make any sense. So w has to be greater than/equal to 0. Statement 2 clears the air by stating w>0. So w is not 0. Hence n not equal to 1. n can still end in 4 0r 6. So we strike out option B
whenever w is odd unit digit is 4...and whenever w is even unit digit is 6 At this point we can say that solution is 6 and strike out options A and D
We needed both satements to arrive at this answer, E can be striked out Hence C is correct.

**(4^1=4,4^2=16......whenever the digit 6 is multiplied by 4...the result will have a number ending in 4(4x6=24).....which in turn when multiplie by 4 again ends in 6(4x4=16)...and so on....)=>

4^w = n, what is the units digit of n?

Considering (i) When w is even integer, it can be ...,-4,-2,0,2,4,......
When w=0, 4^0 = 1, units digit is 1
When w=2, 4^2 = 16, units digit is 6
When w=-2, 4^-2 = 1/16=0.0625, units digit is 0.

We do not get a unique solution. Hence (i) is insufficient

Considering (ii) When w > 0, it can be 1,2,3,......
Also, we are not sure if it is an integer.
When w=1, 4^1 = 4, units digit is 4
When w=2, 4^2 = 16, units digit is 6
When w=3, 4^3 = 64, units digit is 4.

We do not get a unique solution. Hence (ii) is insufficient

Considering (i) and (ii), we get w > 0 and w is an even integer.
So, it can be 2,4,6,8,.....
When w=2, 4^2 = 16, units digit is 6
When w=4, 4^4 = 256, units digit is 6

And the units digit will always be 6, as we multiply all the results by 4^2= 16 and
any two numbers, whose units digit is 6, are multiplied will give a number with 6 as its unit digit.

We do get a unique solution. Hence answer is C.

---------------
ISH

If ,4^w=n what is the units digit of n?

- Rules, if w = 0 Units of N=1, if w<0 Units of n = 0, if w>0 and odd, N = 4 (4,64,1024...), if w>0 and even n = 6 (16,256...)

1. w is an even integer - Insufficient, Units of N = 0 or 6
2. w > 0 - Insufficient Units of N = 4 or 6

Both together, w is even integer, and w>0 n= 6

C - Both together

Correct Answer is Option C.

Option A says that w is an even integer. so w can be 0 also. If w is 0, the unit's digit will be 1. If w is a positive integer greater than 0, then unit's digit will be 6 and if w is a negative integer, then it will be a fraction. So option A alone is not sufficient.

Option B says that w>0, but doesn't specify whether it is an integer or not. So for different values of w, we will get different unit's digits.

Combining both A and B, w is an even integer greater than 0. so unit's place will always be 6.

So C is the answer.

C.

1 - W is Even
If W = 2, units digit of N = 6 (for every even integer).
However, if W=(-2) the units digit of N will be "0"
therefore - not sufficient.

2 - w>0
Of course not sufficient, for example:
W=1/2 - N units digit will be 2
W=1 - Units digit is 4
NOT SUFFICIENT.

(1)+(2)

We know W is positive and even and integer.
so the last digit will always be 6.
For example W=2, N=16. W=4, N = 256

If 4^w=n , what is the units digit of n?

1. 4^2 = 16 4^4 = 256, 4^6 = 4096 => pattern - units digit is "6" - Sufficient
2. 4^1/2 = 2, 4^1 = 4, 4^2 = 16, 4^3 = 64 => different outcomes - Insufficient

A is the answer.

\(4^1=4\)
\(4^2=16\)
\(4^3=64\)
\(4^4=256\)

Conclusion 1 \(4^e\)=units digit of 6 where e is even
Conclusion 2 \(4^o=\)units digit of 4 where o is odd

1) is insufficient since n can have 2 values "6" or "0" because 0 is an even integer and \(4^0=1\)
2) is insufficient because "n" can have 2 values "6" or "4"

Together they are sufficient as it will result in only 1 value i.e. "6"
Ans= C

Before we even look at the statements, the sentence doesn't tell whether it is a positive exponent or if it is an integer, so we have to keep that in mind.
Now, seeing statement 1, tells us that it is an integer indeed, but we still know nothing about its sign. If w < 0 then the units digit is 0.
If w > 0 then the units digit is 6.
Hence NOT SUFFICIENT. We keep answers BCE.

Statement 2 tells us that w is greater than zero, but it fails to address whether it is an integer or not. Therefore NOT SUFFICIENT. We keep answers CE.

Using both statements together we know exactly what we need to know. The coefficient w is greater than zero AND is an integer. We could conclude that the units digit is 6 (we don't have to, though).

The answer is C.

Given in question - 4^w=n
Asked in question - units digit of n

First statement : w is even integer.
As we can see:
4^1 =4---- units digit will be 4
4^2=16--- units digit will be 6
4^3=64--- units digit will be 4
So it can be seen that for all even integer except 0 (0 is even integer) units digit will be 6. And when w=0 the units digit will be 1
Possible values of units digit: 1,6
So STATEMENT 1 is not sufficient

Statement 2 : w>0
When w>0 then we dont know whether w is integer or not. So no fixed unit digit.

So STATEMENT 2 is not sufficient.

But when we combine both statement we get `w` is even integer greater than 0.
Therefore the units digit is 6.

ANSWER : C

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C) Both together

4 to any power can either have a units digit of 4 or 6. If its positive and odd it'll be 4 if its even it'll be six. If its less than zero it'll be a decimal