How many factors of 24 * 53 * 74 are odd numbers?

Key idea: any factor of 2^4 * 5^3 * 7^4 has the form 2^a * 5^b * 7^c with 0 ≤ a ≤ 4, 0 ≤ b ≤ 3, 0 ≤ c ≤ 4.

For a factor to be odd, the exponent of 2 must be 0 (so a = 0).

• Choices for the exponent of 5: b = 0, 1, 2, 3 → 4 choices

• Choices for the exponent of 7: c = 0, 1, 2, 3, 4 → 5 choices

Total number of odd factors = 4 * 5 = 20.

Answer: 20 odd factors.