UPSC Prelims 2026 CSAT, question 68

Topic: Number system

Q68

There are three types of rectangular tiles : $3' \times 3'$ , $3' \times 7'$ and $3' \times 11'$ . An area of rectangular shape of dimensions $3' \times 100'$ is to be covered using these tiles without breaking them. If $x$ and $y$ are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then $x-y$ is

(a) 20Correct answer
(b) 12
(c) 10
(d) 7

Solution.

Answer: (a) 20

Covering a $3' \times 100'$ strip with $3' \times 3'$ , $3' \times 7'$ , and $3' \times 11'$ tiles is equivalent to solving $3a + 7b + 11c = 100$ for non-negative integers $a, b, c$ and finding the maximum and minimum values of $a + b + c$ .
To maximize the number of tiles $x$ , maximize $a$ : $100 = 3(31) + 7(1) + 11(0)$ gives $x = 31 + 1 + 0 = 32$ tiles.
To minimize the number of tiles $y$ , maximize $c$ : $100 = 3(0) + 7(8) + 11(4)$ (or $3(1) + 7(6) + 11(5)$ ) gives $y = 12$ tiles, so $x - y = 32 - 12 = 20$ .