A number for which the sum of all its factors is equal to twice the number is called a ____ number.
prime
Step 1: Recall what factors are.
Factors of a number n are the numbers that divide n exactly.
Step 2: The statement in the question says:
\[ \text{sum of all factors of } n = 2n \]
Step 3: If we subtract the number itself from the sum of all its factors, we get the sum of its proper factors (all factors except the number).
\[ (\text{sum of all factors}) - n = 2n - n \]
\[ \Rightarrow \text{sum of proper factors} = n \]
Step 4: A number whose proper factors add up to the number itself is called a perfect number.
Example 1: 6
Factors: 1, 2, 3, 6
\[ 1 + 2 + 3 + 6 = 12 \]
\[ 2 \times 6 = 12 \]
So, 6 is perfect.
Example 2: 28
Factors: 1, 2, 4, 7, 14, 28
\[ 1 + 2 + 4 + 7 + 14 + 28 = 56 \]
\[ 2 \times 28 = 56 \]
So, 28 is perfect.
Step 5 (Why not prime?): A prime number p has only two factors: 1 and p.
\[ 1 + p = p + 1 \]
This is not equal to \( 2p \) for any \( p > 1 \). So primes do not satisfy the condition.
Conclusion: The blank should be filled with perfect.