UPSC Prelims 2025 CSAT, question 35

Solution.

Answer: (b) 4

1. Since $35 = 5 \times 7$, the highest power $35^n$ dividing the product is $\min(v_5, v_7)$, where $v_5$ and $v_7$ are the exponents of $5$ and $7$ in the prime factorization.
2. Counting factors of $5$: $385 = 5 \times 7 \times 11$ contributes $5^1$ and $1000 = 2^3 \times 5^3$ contributes $5^3$, while $7, 343, 2401,$ and $77777$ have no factor of $5$, giving $v_5 = 1 + 3 = 4$.
3. Counting factors of $7$: $7^1 \times 343 (7^3) \times 385 (7^1) \times 2401 (7^4) \times 77777 (7^1)$ gives $v_7 = 10$; hence $n = \min(4, 10) = 4$.