Define [n] as the greatest integer less than or equal to n.
Given that x is a positive integer, determine analytically all possible solutions to each of the following equations:
(a) [x/3]+ [x/5] + [x/7] = 66
(b)[x/7]+ [x/11] + [x/13] = 245
I'm not sure how analytic a solution Sengupta had in mind. The
following analytically narrows down the possible solution set, and then
tries some values to get a solution.
a) the [x] function subtracts the fractional part of x, which is
an amount that is greater than or equal to 0 and less than 1, for
all x
b) therefore, [x] <= x < ([x] + 1)
c) therefore, 66 <= x/3 + x/5 + x/7 < 69
d) therefore 97.61 < x < 102.04
Having narrowed the range, it is easy to determine that if f(x) = [x/3] + [x/5] + [x/7], then
f(98) = 65
f(99) = 66
f(100) = 67,
f(x) is non-decreasing, so 99 is the only solution to part a.
A similar approach will do part b easily.
Edited on March 25, 2007, 8:36 pm
How analytic is analytic? (spoiler, part a)
