35. 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?

1. A. 6
2. B. 7
3. C. 8
4. D. 9

Answer: C

Explanation

The problem asks for natural numbers N less than 50 such that N = p + q + r, where p, q, and r are distinct factors of N (excluding N itself).
Let’s test numbers systematically:
* The smallest possible sum of three distinct factors (1, 2, 3) is 6. For N=6, factors are 1, 2, 3, 6. Here, 1+2+3 = 6. So, 6 is one such number.
* If N is a multiple of 6, it often has many factors.
* For N=12, factors are 1, 2, 3, 4, 6, 12. We can find 2+4+6 = 12. So, 12 is a number.
* For N=18, factors are 1, 2, 3, 6, 9, 18. We can find 3+6+9 = 18. So, 18 is a number.
* For N=24, factors are 1, 2, 3, 4, 6, 8, 12, 24. We can find 4+8+12 = 24. So, 24 is a number.
* For N=30, factors are 1, 2, 3, 5, 6, 10, 15, 30. We can find 5+10+15 = 30. So, 30 is a number.
* For N=36, factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. We can find 6+12+18 = 36. So, 36 is a number.
* For N=42, factors are 1, 2, 3, 6, 7, 14, 21, 42. We can find 7+14+21 = 42. So, 42 is a number.
* For N=48, factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. We can find 8+16+24 = 48. So, 48 is a number.
The numbers are 6, 12, 18, 24, 30, 36, 42, 48. There are 8 such numbers. This question tests basic number theory and systematic checking, which is common in CSAT.