If R is the set of all numbers that can be written as

Question

If R is the set of all numbers that can be written as 3^6*20^(k+5) *24^(1-k) where k is an integer, what is the lowest whole number power of 10 that is not a factor of any element in R?

(A) 10^9 (B) 10^10 (C) 10^11 (D) 10^12 (E) 10^13

FN · Answer

weird question....

let me look at it this way:

3^6 * 20^k * 20^5 * 24/24^k

so if K=0, then basically we have 3^6 * 20^5 if I approximate that is =20^5

which is 2^5*10^5


so i think the answer shud be 10^5...am I missing something here?

FN · Answer

umm..some amongst the answer choices that would be (a)...

mikki0000 · Answer

I got (B).

This is how I worked it:
Split up the question into its factors based on what we are looking for.

i.e. 3^6 * 2^(k+5) * 10^ (k+5) * 3^ (1-k) * 2 ^(3-3k)

Now basically to arrive at test numbers to plug in for k, I tried to see where they would start to cancel out, leaving the exponent of 10 alone.

so got these numbers by equalizing the exponents of 2 -
k+5 = - (3+3k)
gives k = 4.

In this case what you have is
3^6 * 2^9 * 3 ^ -3 * 2^-9

Now at this point, the factors cancel out, leaving no denominator, and 10^9 as a factor.

If k goes any higher, i.e. 5, you get to a case where there are other denominators that will have to divide the exponent of 10, hence 10^10 cannot be a factor. 

E.g. 3^6 * 2^10 * 3^-4 * 2^-12.
This will leave you with a "4" in the denominator which will have to divide the exponent of 10, leaving you with 10^8 * 5^10. 

The lowest number that is NOT a factor of this set of numbers is therefore 10^10.

MG
MG

FN · Answer

I actually disagree.

the question says lowest power of 10...which would be the case when K is smallest...isnt it?

I mean..if K=0

then the equation is 3^6 * 20^5, which means 2^5*10^5...

here 10^5 is the smallest??? what am I missing here?

kevincan · Answer

If 10^5 is a factor of an element of the set, then 10^5 isn't what we are looking for!

kevincan · Answer

We're looking for the lowest power of 10 that is not a factor of any element.

FN · Answer

OH ****...I am getting really bad at this...didnt read the NOT a factor parrt...hahah

now i follow...

Futuristic · Answer

This is an easier one, considering its from Kevin. Still took way over 2 minutes though

The expression can be reduced to: 3^(7-k). 2^(13-k). 2(k+5). We need to find out the highest power of 10 that IS supported. One higher than this power is the lowest power of 10 that is NOT supported. Clearly we're looking for (2.5)^x. for k=0 power of 2 = 13, power of 5 = 5 k=1 2: 12, 5:6 let k =4 we get power of 2 =9, power of 5 = 9. This is clearly the optimal value. If we increase it, we form one less power of 10. If we decrease it we get one less power of 10. 10^9 is therefore the highest factor. 10^10 is therefore the lowest power of 10 which is not a factor.

If R is the set of all numbers that can be written as

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