Q5. What is the maximum value of n such that 7 × 343 × 385 × 1000 × 2401 × 77777 is divisible by 35^n ?

To find the maximum value of 'n' for which the given expression is divisible by 35^n, we first express 35 in its prime factors: 35 = 5 × 7. So, 35^n = 5^n × 7^n. We need to find the total powers of 5 and 7 in the given expression: 1.

7 = 7^1 2. 343 = 7^3 3. 385 = 5 × 7 × 11 4. 1000 = 10^3 = (2 × 5)^3 = 2^3 × 5^3 5. 2401 = 7^4 6. 77777 = 7 × 11111 (11111 is not divisible by 5 or 7) Combining the powers of 5: 5^1 (from 385) + 5^3 (from 1000) = 5^(1+3) = 5^4.

Combining the powers of 7: 7^1 (from 7) + 7^3 (from 343) + 7^1 (from 385) + 7^4 (from 2401) + 7^1 (from 77777) = 7^(1+3+1+4+1) = 7^10. The expression can be written as 2^3 × 5^4 × 7^10 × 11 × 11111.

For this to be divisible by 5^n × 7^n, 'n' must be less than or equal to the minimum of the powers of 5 and 7. The minimum power is 4 (from 5^4). Therefore, the maximum value of n is 4.

This question tests fundamental concepts of number theory, specifically prime factorization and divisibility rules, which are common in the UPSC CSAT syllabus.