A natural number N is such that it can be expressed as N = p + q + r, where p, q, and r are distinct factors of N. How many numbers below 50 have this property?

Correct Option (3)

A natural number N possesses the property if it can be expressed as the sum of three distinct factors, p, q, and r, such that N = p + q + r. To satisfy this condition, N must have at least three distinct factors.

The smallest possible sum of three distinct factors is 1 + 2 + 3 = 6. Therefore, any number N with this property must be at least 6.

We systematically identify such numbers below 50 (i.e., from 6 to 49):

• N = 6: Factors are {1, 2, 3, 6}. Here, 1 + 2 + 3 = 6.
• N = 12: Factors are {1, 2, 3, 4, 6, 12}. Here, 2 + 4 + 6 = 12.
• N = 18: Factors are {1, 2, 3, 6, 9, 18}. Here, 3 + 6 + 9 = 18.
• N = 24: Factors are {1, 2, 3, 4, 6, 8, 12, 24}. Here, 4 + 8 + 12 = 24.
• N = 30: Factors are {1, 2, 3, 5, 6, 10, 15, 30}. Here, 5 + 10 + 15 = 30.
• N = 36: Factors are {1, 2, 3, 4, 6, 9, 12, 18, 36}. Here, 6 + 12 + 18 = 36.
• N = 42: Factors are {1, 2, 3, 6, 7, 14, 21, 42}. Here, 7 + 14 + 21 = 42.
• N = 48: Factors are {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}. Here, 8 + 16 + 24 = 48.

A total of 8 numbers below 50 satisfy the given property.

Incorrect Options:

Options 1 (6), 2 (7), and 4 (9) are incorrect because a comprehensive enumeration of natural numbers below 50 that can be expressed as the sum of three distinct factors reveals exactly 8 such numbers. These options represent an inaccurate count of the numbers possessing the specified property.