What is the maximum value of n such that 7 × 343 × 385 × 1000 × 2401 × 77777 is divisible by 35ⁿ?

Answer: (b) 4

35ⁿ needs n factors of 5 and n factors of 7. The product has only four 5s (one from 385, three from 1000) but ten 7s, so the maximum n is 4.

1. 35 = 5 × 7, so count the 5s and the 7s in the product; n is the smaller count.
2. Fives: 385 = 5 × 7 × 11 gives one, and 1000 = 2³ × 5³ gives three; 7, 343, 2401 and 77777 have none. Total = 4.
3. Sevens: 7 (one), 343 = 7³ (three), 385 (one), 2401 = 7⁴ (four), 77777 = 7 × 11111 (one). Total = 10.
4. Each 35 needs one 5 and one 7, so n = the smaller of 4 and 10 = 4.

Remember · For divisibility by a power of a composite number, count each prime factor separately; the scarcer prime sets the limit.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error