The number of even divisors of 100 will be

Concept

If a positive integer has prime factorization n = p1a₁ × p2a₂ × ⋯, every positive divisor is formed by independently choosing the exponent of each prime from 0 up to its exponent in n.

To count divisors with an extra condition, restrict the allowed exponent choices first, then multiply the numbers of independent choices.

Application

1. Write the prime factorization: 100 = 22 × 52.

2. An even divisor must contain a factor of 2, so the exponent of 2 can be 1 or 2. This gives 2 choices.

3. The exponent of 5 can be 0, 1, or 2. This gives 3 choices.

4. By the multiplication principle, the number of even divisors is 2 × 3 = 6.

Cross-check

The complete list of even divisors is:

• 2

• 4

• 10

• 20

• 50

• 100

Each listed number divides 100 and is even, and the list contains exactly six values.

Result

Therefore, the number of even divisors of 100 is 6.