If x is a positive integer, what is the hundreds digit of x^x ?

Question

(1) x is divisible by 10.
(2) x is even.

M13-36

amanvermagmat · Answer

(1) The minimum possible value of x is 10, and in this case 10^10 = 10000000000. Clearly the hunderds digit is '0'. If x=20, then 20^20 will end in more zeroes than 10^10, so definitely here also x^x would have a hunderds digit of '0'. Whether x be 30 or 40 ... hundreds digit will be '0' only. 
So if x is a positive integer which is a multiple of 10, hundreds digit will be '0' only. Sufficient.

(2) 4^4 is 256, here hundreds digit is '2'; but 10^10 has a hundreds digit of '0'. Different answers for different values of x (both 10 and 4 are even). 
Not sufficient.

Hence  A answer

EgmatQuantExpert · Answer

Solution

Given:  
• x is a positive integer

To find:  
• The hundreds digit of  x^x

Analysing Statement 1  
• As per the information given in Statement 1, x is divisible by 10
 o If we assume the lowest possible value of x, i.e. 10, then  10^{10}  will have 10 zeroes from the right before the first non-zero digits.
o Higher multiples of 10 will also have at least 10 zeroes before the first non-zero digits

Hence, from statement 1, we can answer the question

Analysing Statement 2  
• As per the information given in Statement 1, x is even
 o x can be any even number, and depending on the value of x, the hundreds digit of  x^x  will differ

Hence, statement 2 is not sufficient to answer

Hence, the correct answer is option A.

Answer: A

mohitagarwal · Answer

Considering statement 1: x is divisible by 10; can x be equal to 0 as "0 is divisible by any number"?
If x=0, then statement 1 is not sufficient as  0^0  = 1 and we don't have a hundreds digit here.

IMO   E   would be the answer. Please correct me if I am wrong.

EgmatQuantExpert · Answer

In the question it is mentioned that x is a positive integer. Hence, x cannot be equal to 0.

ccooley · Answer

0^0  actually isn't used on the GMAT at all, since mathematicians haven't agreed on a single set value for it. (Every other number raised to the 0 power is definitely equal to 1, but  0^0  is a special case.) So, if you see numbers raised to the 0 power in a GMAT problem, the problem should always specify that the base is not equal to 0.

This problem achieves that by specifying that the base is positive. Zero is neither positive nor negative, so the base can't equal zero.

Bunuel · Answer

Official Solution:

If  x  is a positive integer, what is the hundreds digit of  x^x ? 
 
(1)  x  is divisible by 10.
 
This implies that  x  could be 10, 20, 30, and so on. Thus,  x^x  could be  10^{10} ,  20^{20} ,  30^{30} , etc. The hundreds digit of any of these numbers will be 0, as each of these numbers ends with at least ten zeros (because they have factors of both  5^{10}  and  2^{10} ). Sufficient.
 
(2)  x  is even.
 
For  x=4 ,  4^4 = 256 , so the hundreds digit is 2. However, for  x=10 ,  10^{10}=10,000,000,000 , so the hundreds digit is 0. Not sufficient.

Answer: A

If x is a positive integer, what is the hundreds digit of x^x ?

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If x is a positive integer, what is the hundreds digit of x^x ?

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