Multiple choice general knowledge math & puzzles

How many distinct rectangles of perimeter 258 cm have both length and breadth as positive integers, when expressed in cms

  1. 64

  2. 65

  3. 128

  4. 129

  5. 212

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A Correct answer
Explanation

For perimeter 258, if length = l and breadth = b, then 2(l+b) = 258, so l+b = 129. For integer dimensions, l can be 1 to 128, giving 128 ordered pairs. Since l>l+b, b>0 gives l<129, and we need l >= b for distinct rectangles. Counting gives 32 values of l from 65 to 96, and 32 values where l < b, totaling 64.