Bases Trouble a Lot!
Let x and y be positive integers such that ( 3 4 ) x = ( 4 3 ) y . Determine the 2 0 1 4 th minimum value of x − y .
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1 solution
Isn't there any other way to do it? Is this you own question-If yes, the n awesome!!!
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@Krishna Ar Thanks. I don't know any other way. This is an 'entirely-mine' problem. BTW.........What is your calculus rating at present. Check it.. :D
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1767 I totally don't deserve it. Yours?
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@Krishna Ar – 1500 I totally don't deserve it. How d'you know calculus? We don't have it in CBSE any before 12th.
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@Satvik Golechha – Yes. That's a very sorry state our syllabus is in. I actually only have STARTED learning it. You? And, I am facing a big dilemma!!! :( To do or not to do- Calculus vs. Olympiad prep?!!!. How are you going to do? What are your plans to prep?..... @Satvik Golechha
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@Krishna Ar – Wait! @Krishna Ar You say that you've just started learning it........... and you're a level 4. I can't believe it. BTW I need to learn basic calculus for my physics.
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@Satvik Golechha – @Satvik Golechha -Uh..Uhm!!!! THERE are a LOT of easy calculus problems rated 2000 and above which helped me get there....I totally dont deserve them as much as I deserve the other ratings!!!! Prep going on for math oly???
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@Krishna Ar – Nah! Brilliant doesn't give me any time for anything but school stuff. @Krishna Ar Will you mind telling me those LOT of easy calculus problems rated 2000 and above?
My solution uses similar methods to yours though through a different rearrangement of equations:
Using your given equation were 3 x + 4 = 4 y + 3 , we can rearrange to get:
1 = 4 y − 3 x = y − 3 ( x − y )
From our equations we can see that we cannot have y > x as x , y ∈ N so we must have x − y ≥ 0 . But for x − y = 0 , 1 we have y = 1 , 4 which is inadmissable for a base that has 4 as a digit. So the minimum value for x − y is thus 2, and from there we can see that it can take on any integer values ≥ 2 . Hence the 2014th minimum value of x − y is then 2 0 1 5
Oops.. I just forgot to rule out those two solutions..
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That's why its at lvl#4...
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U leveled up in geo too!!!! wow!!!!
Let us convert the given equation in base 10. 3 x + 4 = 4 y + 3 or x = 3 4 y − 1 Now, we see that the solutions are ( 1 , 1 ) , ( 5 , 4 ) ,....., ( 4 n − 3 , 3 n − 2 ) ,..... But the first two solutions are not correct since we can't have 4 3 in base 1 or 4. So now we make a sequence of the differences of x and y . It starts from ( 9 , 7 ) , so the series is 2 , 3 , 4 , 5 , . . , n + 1 , . . Thus, the 2 0 1 4 t h difference is 2 0 1 5