How many factors of $$2^7×3^4×5^3×7$$​ are even?

Correct option is D

Given:

Number: 2⁷ × 3⁴ × 5³ × 7

Solution:

The number is already in prime factorized form: 2⁷ × 3⁴ × 5³ × 7¹

Increase each exponent by 1 and multiply:

(7 + 1) × (4 + 1) × (3 + 1) × (1 + 1) = 8 × 5 × 4 × 2 = 320

Consider only the odd factors excluding 2:

3⁴ × 5³ × 7¹

Increase each exponent by 1 and multiply:

(4 + 1) × (3 + 1) × (1 + 1) = 5 × 4 × 2 = 40

Subtract the number of odd factors from the total number of factors:

320 (total factors) - 40 (odd factors) = 280

Therefore, there are 280 even factors of 2⁷ × 3⁴ × 5³ × 7.