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Integer Equations (Posted on 2007-03-25) Difficulty: 2 of 5
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

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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How analytic is analytic? (spoiler, part a) Comment 2 of 2 |
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
  Posted by Steve Herman on 2007-03-25 20:34:29

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