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Question
Evaluate:
Σ[log k] over all k
where k is a factor of 1 000 000 000 000
except 1 and 1 000 000 000 000 it self
([x] is the greatest integer function)
... ...
Hey Lawrence~
)
I will make a few comments that might be helpful:
The greatest integer function is just the largest integer <= to a real number, i.e., [2.5] = 2 because 2 is the last integer before 2.5. Also when dealing with negative numbers the greatest integer function uses the same rule, for example [-1.2] = ? What integer is less than or equal to -1.2 but there is no integer between it and -1.2? Yes -2.
It appears you need to sum the greatest integer function of the log function where k is a factor of one trillion. There are many powers of 10 that operate as factors of one trillion but there are other factors as well. One way to narrow this down is to prime factor one trillion. It will be composed of 2's and 5's, namely 2^12*5^12, so various combination's of those powers of 2 and 5 are all candidates to be factors of one trillion. You could grunt it out and figure how many different ways to get factors in this way. Take 2 and pair it (as factors) with 5, 5^2, 5^3,...,5^12 and then do it again with 2^2, 2^3,...,2^12 and you will see that there are 12^2 ways to find factors of one trillion or 144 different factors of one trillion. BUT in your instructions it says to not include 1 or one trillion. We didn't use 2^0 or 5^0 both equal to 1 so we don't have to worry about either of those being one of the 144 but we did couple 2^12 with 5^12 which is one billion so we need to eliminate that one combination so we really have 143 k's to try. [Note: there is only one combination composed of two factors that equals 1 trillion and no pairs of factors whose product equal 1, that is why there are 144-1 = 143 combinations to sum the greatest integer function from.] So you want to sum all those greatest integers with those 144 k numbers. I can promise you that every power of 2 by itself or every power of 5 by itself does not generate an integer so for every power of two you would need to determine the value of the log and find that values greatest integer. The good news is that all powers of 10 are integers already, log 10 = 1, log 100 =2, log 1000 = 3, log 10,000 = 4,,0...,log(100,000,000,000)(100 billion = 11 zeros)= 11, do you see the pattern? Remember we won't be looking at log(12 zeros = 1 trillion) so we stopped at 100 billion).
Another interesting observation is that when calculating the greatest integer function on the powers of 2 there is a pattern: 0+0+0+1+1+1+2+2+2+3+3+3 = 18 and the powers of 5: 0+1+2+2+3+4+4+5+6+6+7+8 = 48. Also the sum of the powers of 10 are equal to the sums of the powers of 2 and the powers of 5 for a grand total of 66+66 = 132, but you need to crunch it to make sure I haven't made any mistakes since I did this all by hand. Also you will need to figure the other 143-36 = 107 cases that I didn't hand calculate for you. I'm going to attach an image of a table that you could duplicate to make it easier to track your info.
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Questioner's Rating
| Rating(1-10) | Knowledgeability = 10 | Clarity of Response = 9 | Politeness = 9 |
| Comment | hey... i cant understand your solution but i appreciate your respondse , thanks... u said that the number of ways of combination is 143 but i think it is 142 because 12^2-1-1 = 144-2=142... thanks | ||
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Sherry Wallin
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