Question number 27. The equalities k is equal to quantity 2014 plus m raised to the power 1 over n equal to 1024 raised to the power 1 over n plus 1 are given for positive integers k, m, and n. How many different values can the number m take? So let's begin by taking these two at a time and solving for m and solving for k. So what we can do is we can say k is first of all equal to 2014 plus m raised to the power 1 over n. We can raise both sides to the power n so k to the power n is equal to 2014 plus m and then we finally have that m is equal to k to the power n minus 2014 and this is a positive integer. So based on the values of k and n if we obtain a negative number here we will discount that possibility for m. So now let's look at k in terms of m and n. Let's look at the second equation now k is equal to 1024 raised to the power n plus 1 so we can solve for n this time k minus 1 is 1024 to the power 1 over n and so raising both sides to the power n we see that k minus 1 quantity raised to the power n is 1024 which we should recognize as 2 to the power 10 and then what we know is since k minus 1 is an integer we can see that k minus 1 raised to the power n equals 2 to the power 10 implies that in fact k to the power excuse me k minus 1 is going to have to be a power of 2 so let's say 2 to the power p for some positive integers p and then if that's so then we look at 2 to the power 10 now that's equal to 2 to the power p raised to the power n and we see that 10 is the product of the positive integers p and n so that can't happen very many in very many situations we can in fact then tabulate all the possibilities so let's let's draw a table here in the first column i will have the values of of p in the second column i will have values of n and i can solve for p in terms of n or n in terms of p so let's say that n is going to be 10 divided by p and what can happen so if p is equal to 1 the smallest positive integer then n is equal to 10 if p is equal to 2 then n is equal to 5 and then since n has to be a positive integer i only have uh one or two more choices here if p is equal to 5 n is 2 and finally if p is equal to 10 n is equal to 1 and those are my only choices since i'm forced to work with positive integers so then with that let's uh let's see what the value of um k is in the next column i will have here k and we have solved for k already k is equal to 1024 to the power 1 over n plus 1 and if n is equal to 10 then this is exactly equal to 2 to the power 10 over 10 plus 1 so that's that's equal to 3 if if n is equal to 5 then we have 2 to the power 10 over 5 so 2 plus 1 so that gives us 5 if n is equal to 2 then we're looking at 2 to the power 5 plus 1 so that is 33 and if n is equal to 1 then we have 2 to the power 10 plus 1 so that is 1025 and so finally with values of k we can find values of m so let's have one more column over here column over here we have solved for and let's extend this row a little bit so we have solved for m and m was equal to 2 excuse me k to the power n minus 2014 so here if k is 3 and n is 10 then we have these numbers here let's just fill them in then we'll talk about them so k is 5 n is 5 minus 2014 and then k is 33 n is 2 minus 2014 and m is going to be finally equal to k 1025 raised to the power 1 minus 2014 and then looking at these we know that m has to be a positive integer so if we look at 1025 minus 2014 that's a negative number so we can't have that 33 squared is something like on the order of a thousand minus 2014 that's that's going to be also a negative number clearly 2 to the 10 is 2000 is 1024 so 3 to the 10th is going to be greater it's going to be a positive number over here and then what is 5 to the power 5 and if we do that calculation 5 to the power 3 is 125 to the fourth power 625 625 times 5 is going to be greater than 2014 so this is also a positive number so only two possibilities um for m if we demand that m is a positive integer and the answer then is going to be in that case c 2

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