Correct Option (B)
To determine the maximum value of n such that the given product is divisible by 35n, we first express the divisor 35 in its prime factors. Since 35 = 5 × 7, it follows that 35n = 5n × 7n. For the product to be divisible by 35n, it must contain at least n factors of 5 and n factors of 7.
We proceed by performing the prime factorization for each term in the given product:
- 7 = 71
- 343 = 73
- 385 = 51 × 71 × 111
- 1000 = 23 × 53
- 2401 = 74
- 77777 = 71 × 411 × 2711
Next, we calculate the total count of each prime factor (5 and 7) across all terms in the product:
- Total power of 5: The factors of 5 appear in 385 (51) and 1000 (53). The sum of their exponents is 1 + 3 = 4.
- Total power of 7: The factors of 7 appear in 7 (71), 343 (73), 385 (71), 2401 (74), and 77777 (71). The sum of their exponents is 1 + 3 + 1 + 4 + 1 = 10.
For the product to be divisible by 5n × 7n, 'n' must be less than or equal to the minimum of the total powers of 5 and 7 available in the product. Therefore, n = min(4, 10) = 4.
The maximum value of n is 4.
Incorrect Options:
The maximum value of n is constrained by the prime factor with the lowest total power, which is 5 with a total power of 4. Consequently, any value of n greater than 4 (such as 5 or 7, corresponding to options C and D) would imply that the product must contain more factors of 5 than are actually present (only 4 factors of 5 are available). Option A (3) is incorrect because while the product is divisible by 353, the question specifically asks for the maximum possible value for n, which is 4.