UPSC CDS 1 2020 Math Paper

Concept:If a divisor leaves the same remainder when dividing multiple numbers, then the divisor must also divide the differences of those numbers.Explanation:Let the required number be $d$ and the common remainder be $r$.Then $(12288 - r)$, $(28200 - r)$, and $(44333 - r)$ are exactly divisible by $d$.The difference of any two such numbers is also divisible by $d$.Calculate the differences:$(28200 - r) - (12288 - r) = 28200 - 12288 = 15912$$(44333 - r) - (28200 - r) = 44333 - 28200 = 16133$Thus $d$ must be a divisor of both $15912$ and $16133$.Find the highest common factor (HCF) of these two numbers.Factorize $15912$: $15912 = 2^3 \times 3^2 \times 13 \times 17$Factorize $16133$: $16133 = 13 \times 17 \times 73$Common prime factors: $13$ and $17$.HCF = $13 \times 17 = 221$.Therefore, the largest number satisfying the condition is $221$.Answer:$221$ (Option C).