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Let a and b be positive integers such that- If the minimum value of b is , find n.
(The question is a modified form of one which appeared in INMO 2005.)
The answer is 5.
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1 solution
Inverse the whole relation and write
77/17 < b/a < 197/43
ie. 4 + 9/17 < b/a < 4 + 25/43
ie. 4 < b/a < 5
Since b and a are positive integers
b can be written as b = 4a + n ............. (1)
ie. b/a = 4 + n/a ie. 9/17 < n/a < 25/43
ie. 43n/25 < a < 17n/9
Let n = 1, 2, 3, 4
When you plug in n = 4 we see that
6 + 22/25 < a < 7+ 5/9
This means that the maximum value a can have is 7
But b = 4a + n
So b = 4*7 + 4 = 28 + 4 = 32
and 2^5 = 32.