Number System Question 5 - CAT 2025

Step-by-Step Explanation

• Establish structural divisor rules:

Every factor of the expression must take the general form 2ᵃ × 3ᵇ × 5ᶜ × 7ᵈ, where the exponents are bounded by 0 ≤ a ≤ 6, 0 ≤ b ≤ 5, 0 ≤ c ≤ 3, and 0 ≤ d ≤ 2.

For a divisor to match the form 3r + 1, it cannot be a multiple of 3. This forces b = 0.

• Apply modulo arithmetic:

Evaluate the remaining prime bases modulo 3:

2 ≡ - 1 (mod 3), 5 ≡ - 1 (mod 3), 7 ≡ 1 (mod 3)

Divisor ≡ ( - 1)ᵃ × ( - 1)ᶜ × (1)ᵈ ≡ ( - 1)ᵃ⁺ᶜ (mod 3)

For the result to equal 1 (mod 3), the combined exponent a + c must be an even integer.

• Count the matching exponent pairs:

An even sum requires both exponents to be either even or odd.

• a in 0,1,2,3,4,5,6 (4 even choices, 3 odd choices)
• c in 0,1,2,3 (2 even choices, 2 odd choices)

Valid combinations for (a + c) = ( even × even ) + ( odd × odd )

Combinations = (4 × 2) + (3 × 2) = 8 + 6 = 14 pairs

For every valid pair, the exponent d can be chosen in 3 independent ways (0,1,2).

Total valid divisors = 14 × 3 = 42

Correct Option: D