UPSC CDS II 2024 Math Paper

Concept:Find a number that is a multiple of 11 and leaves remainder 5 when divided by 6, 12, 15, 18. Use LCM and modular arithmetic.Explanation:First, find the least common multiple (LCM) of 6, 12, 15, 18.Prime factorize: $6=2\times3$, $12=2^2\times3$, $15=3\times5$, $18=2\times3^2$.LCM = $2^2\times3^2\times5 = 4\times9\times5 = 180$.The number $N$ satisfies: $N \equiv 5 \pmod{180}$, so $N = 180K + 5$ for some integer $K \ge 0$.Also, $N$ must be a multiple of 11: $180K + 5$ is divisible by 11.Check values of $K$ from 0 upward until divisible by 11.$K=0$: $180\times0+5=5$ (not divisible by 11).$K=1$: $185$, $185\div11=16.81$ (not integer).$K=2$: $365$, $365\div11=33.18$.$K=3$: $545$, not integer.$K=4$: $725$, not.$K=5$: $905$, not.$K=6$: $1085$, $1085\div11=98.636$.$K=7$: $1265$, $1265\div11=115$ (exact).Thus the least $K$ that works is $K=7$, giving $N=1265$.Verify: $1265$ remainder when divided by 6 is $5$? $1265 \div 6 = 210 \times 6 = 1260$, remainder $5$. Similarly for 12, 15, 18 gives remainder 5. So $N=1265$ is correct.Now $1265$ lies between $1200$ and $1300$.Answer:$1200 < N < 1300$, so option D is correct.