What is the greatest number of rectangles with integer side lengths and perimeter 10 that can be cut from a piece of paper with width 24 and length 60?

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Answer 1 · 2016-08-22T20:20:58

$360$

If a rectangle has perimeter $10$ then the sum of its length and width is $5$ , giving two choices with integer sides:

The piece of paper has area $24xx60 = 1440$

This can be divided into $12xx20=240$ rectangles with sides $2xx3$ .

It can be divided into $24xx15 = 360$ rectangles with sides $1xx4$

So the greatest number of rectangles is $360$ .

Answer 2 · 2016-08-23T09:02:41

$360$

Calling $S = 60 xx 24 = 2^5 xx 3^2 xx 5 xx 1$ the problem can be stated as

Determine

$max n in NN^+$

such that

$n le S/(a cdot b)$
$a + b = 5$
${a, b} in {1,2,3,4}$

giving the feasible pairs

${1,4},{2,3}$ and the desired result is

$n = 1440/4 =360$