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

Explanation : 

We need the largest integer n such that the product

P = 7 × 343 × 385 × 1000 × 2401 × 77777

is divisible by 35ⁿ. Since 35 = 5·7, 35ⁿ = 5ⁿ·7ⁿ, so n cannot exceed the exponent of 5 in P or the exponent of 7 in P. We compute those exponents term by term.

1. Factor each term by 5 and 7:

• 7 = 7¹
• 343 = 7³
• 385 = 5¹·7¹·11
• 1000 = 2³·5³
• 2401 = 7⁴
• 77777 = 7·11111 = 7¹·41·271

2. Total exponent of 7 in P:

v₇(P) = 1 (from 7)
+ 3 (from 343)
+ 1 (from 385)
+ 0 (from 1000)
+ 4 (from 2401)
+ 1 (from 77777)
= 10

3. Total exponent of 5 in P:

v₅(P) = 0 (from 7)
+ 0 (from 343)
+ 1 (from 385)
+ 3 (from 1000)
+ 0 (from 2401)
+ 0 (from 77777)
= 4

4. Since 35ⁿ = 5ⁿ·7ⁿ, the maximum n is the smaller of v₅(P) and v₇(P):

nₘₐₓ = min(4, 10) = 4.

Therefore, the product is divisible by 35⁴ but not by 35⁵, so the answer is 4.