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

(a) 3
(b) 4 ✓ UPSC's answer
(c) 5
(d) 7

Why the answer is (b)

• Factorize 35 as 5 × 7, so the number must be divisible by 5ⁿ and 7ⁿ; the maximum n is the minimum of the exponents of 5 and 7 in the prime factorization of the product.

• Calculate the total power of 5: 385 = 5 × 77 (one 5), 1000 = 10³ = 2³ × 5³ (three 5s); total exponent of 5 is 1 + 3 = 4.

• Calculate the total power of 7: 7 = 7¹, 343 = 7³, 385 = 5 × 7 × 11 (one 7), 2401 = 7⁴, 77777 = 7 × 11111 (one 7); total exponent of 7 is 1 + 3 + 1 + 4 + 1 = 10.

• The limiting factor is the power of 5, which is 4, while the power of 7 is 10.

• Therefore, the maximum value of n such that 35ⁿ divides the product is min(4, 10) = 4.

Why the other options are wrong

(a) 3: Option (a) is incorrect because the product contains four factors of 5, allowing for a higher power of 35.
(c) 5: Option (c) is incorrect because there are only four factors of 5 in the product, not five.
(d) 7: Option (d) is incorrect because the number of factors of 5 is only four, which limits the power of 35 to 4.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.