Find the number of divisors of 720. How many of these are even? Also find the sum of divisors.

To find the number of divisors of 720, we first need to determine its prime factorization. The prime factorization of 720 is:

Prime Factorization of 720

720 can be expressed as:

• 720 = 2^4 × 3^2 × 5^1

Calculating the Total Number of Divisors

The formula to find the total number of divisors from the prime factorization is:

• If a number is expressed as p₁^a × p₂^b × p₃^c, then the number of divisors is (a + 1)(b + 1)(c + 1).

Applying this to 720:

• (4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30

Thus, 720 has 30 divisors.

Finding the Even Divisors

To find the even divisors, we can consider the prime factorization again. An even divisor must include at least one factor of 2. The remaining factors can be any combination of the other primes:

• For even divisors, we can have 2¹ to 2⁴ (4 choices), 3⁰ to 3² (3 choices), and 5⁰ to 5¹ (2 choices).

Calculating the number of even divisors:

• (4)(3)(2) = 24

So, there are 24 even divisors of 720.

Sum of Divisors

The sum of the divisors can be calculated using the formula:

• For p₁^a × p₂^b × p₃^c, the sum is:
• (p₁⁰ + p₁¹ + ... + p₁^a)(p₂⁰ + p₂¹ + ... + p₂^b)(p₃⁰ + p₃¹ + ... + p₃^c).

Calculating for 720:

• (1 + 2 + 4 + 8 + 16)(1 + 3 + 9)(1 + 5)
• = 31 × 13 × 6 = 2418

The sum of the divisors of 720 is 2418.

Summary

• Total divisors: 30
• Even divisors: 24
• Sum of divisors: 2418