UPSC CDS 2 2021 Math Paper

Concept:For the product of five consecutive integers to equal one of them, the product must be zero. Hence, zero must be included in the sequence.Explanation:Let the five consecutive integers be $n$, $n+1$, $n+2$, $n+3$, $n+4$.Their product is $P = n(n+1)(n+2)(n+3)(n+4)$.If $P$ equals one of these integers, say $k$, then either $k=0$ (so $P=0$) or the product of the other four numbers is $1$.The product of four consecutive integers cannot be $1$ because consecutive integers are distinct and non‑zero except for cases like $(-1,0,1,2)$, whose product is $0$, not $1$.Thus, the only possibility is that one of the integers is zero.The zero can be at any of the five positions in the sequence of five consecutive integers:• $-4,-3,-2,-1,0$• $-3,-2,-1,0,1$• $-2,-1,0,1,2$• $-1,0,1,2,3$• $0,1,2,3,4$Hence, there are exactly five such sequences.Answer:D. Five