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

Question

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

1. A. 3
2. B. 4 (Correct answer)
3. C. 5
4. D. 7

Correct Answer

Option B — 4

Detailed Solution & Explanation

The correct answer is 4.

Key Points

• 7 × 343 × 385 × 1000 × 2401 × 77777.
• 35 = 5 × 7. Count factors of 5: 385=5×7×11 (one 5), 1000=8×125=5³ (three 5s), 77777=7×11111 (no 5).
• Total 5s = 4. Count factors of 7: 7 (one), 343=7³ (three), 385=7×...
• (one), 2401=7⁴ (four), 77777=7×11111 (one).
• Total 7s = 10. 35ⁿ needs n fives and n sevens.
• Limiting factor is 5s = 4. Maximum n = 4.