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?
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Finding Natural Numbers with a Special Factor Property
The problem asks us to find the count of natural numbers N, less than 50, that can be expressed as the sum of three distinct factors of N. Let these distinct factors be p, q, and r, such that p < q < r and N = p + q + r.
Understanding the Condition N = p + q + r
For N to be the sum of three distinct factors, it must have at least three distinct factors. Numbers with only one or two factors (prime numbers and 1) are immediately ruled out. Also, the smallest possible sum of three distinct factors for any number greater than 1 will always be the sum of its three smallest factors. These are typically 1, the smallest prime factor, and the next smallest factor.
For N = p + q + r, where p, q, r are distinct factors of N, the smallest possible sum of three distinct factors is $1 + (\text{smallest prime factor}) + (\text{second smallest factor})$. Since p, q, and r must be distinct, $p \ge 1$, $q \ge 2$ (if 2 is a factor) or $q \ge 3$ (if 2 is not a factor but 3 is), and $r$ would be the next distinct factor. If N has factors 1, 2, and 3, the smallest sum is $1+2+3=6$. Thus, any number N satisfying this property must be at least 6.
Checking Natural Numbers Below 50
We need to examine each natural number N from 6 up to 49. For each N, we will find all its factors and check if any combination of three distinct factors sums up to N.
Let's check some numbers:
- N = 6: Factors are 1, 2, 3, 6. Can we find three distinct factors p, q, r such that p + q + r = 6? Let's check combinations:
- 1 + 2 + 3 = 6. Yes. So, N=6 satisfies the property.
- N = 12: Factors are 1, 2, 3, 4, 6, 12. Can we find three distinct factors p, q, r such that p + q + r = 12? Let's check combinations:
- Smallest possible sum of three distinct factors: 1 + 2 + 3 = 6 (< 12)
- Larger combinations: 2 + 4 + 6 = 12. Yes. So, N=12 satisfies the property.
- N = 18: Factors are 1, 2, 3, 6, 9, 18. Can we find three distinct factors p, q, r such that p + q + r = 18?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 3 + 6 + 9 = 18. Yes. So, N=18 satisfies the property.
- N = 24: Factors are 1, 2, 3, 4, 6, 8, 12, 24. Can we find three distinct factors p, q, r such that p + q + r = 24?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 4 + 8 + 12 = 24. Yes. So, N=24 satisfies the property.
- N = 30: Factors are 1, 2, 3, 5, 6, 10, 15, 30. Can we find three distinct factors p, q, r such that p + q + r = 30?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 5 + 10 + 15 = 30. Yes. So, N=30 satisfies the property.
- N = 36: Factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Can we find three distinct factors p, q, r such that p + q + r = 36?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 6 + 12 + 18 = 36. Yes. So, N=36 satisfies the property.
- N = 42: Factors are 1, 2, 3, 6, 7, 14, 21, 42. Can we find three distinct factors p, q, r such that p + q + r = 42?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 7 + 14 + 21 = 42. Yes. So, N=42 satisfies the property.
- N = 48: Factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Can we find three distinct factors p, q, r such that p + q + r = 48?
- Smallest sum: 1 + 2 + 3 = 6.
- Try combinations: 8 + 16 + 24 = 48. Yes. So, N=48 satisfies the property.
We continue this process for all numbers from 6 to 49. We found that numbers 6, 12, 18, 24, 30, 36, 42, and 48 satisfy the property.
| Number N | Factors of N | Distinct Factors p, q, r such that N = p + q + r | Satisfies Property? |
|---|---|---|---|
| 6 | 1, 2, 3, 6 | 1 + 2 + 3 = 6 | Yes |
| 12 | 1, 2, 3, 4, 6, 12 | 2 + 4 + 6 = 12 | Yes |
| 18 | 1, 2, 3, 6, 9, 18 | 3 + 6 + 9 = 18 | Yes |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 4 + 8 + 12 = 24 | Yes |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 5 + 10 + 15 = 30 | Yes |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 6 + 12 + 18 = 36 | Yes |
| 42 | 1, 2, 3, 6, 7, 14, 21, 42 | 7 + 14 + 21 = 42 | Yes |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 8 + 16 + 24 = 48 | Yes |
List of Numbers Below 50 with the Property
The natural numbers below 50 that satisfy the property N = p + q + r, where p, q, and r are distinct factors of N, are:
- 6
- 12
- 18
- 24
- 30
- 36
- 42
- 48
Conclusion
By checking each number from 6 to 49, we found that there are 8 such numbers that can be expressed as the sum of three distinct factors.
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Natural Number | A positive integer (1, 2, 3, ...). | The numbers N we are examining are natural numbers below 50. |
| Factor (Divisor) | A number that divides another number evenly (with no remainder). | The numbers p, q, and r must be factors of N. |
| Distinct Factors | Factors that are different from each other (p ≠ q, q ≠ r, p ≠ r). | The condition explicitly states the factors must be distinct. |
| Sum of Factors | Adding the values of selected factors. | We check if N equals the sum of three distinct factors. |
Additional Information: Factor Properties
Exploring factors is a fundamental part of number theory. Every natural number greater than 1 has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers. Numbers with more than two factors are called composite numbers.
Finding factors of a number N involves testing divisibility by numbers from 1 up to $\sqrt{N}$. If a number i divides N evenly, then i and N/i are both factors. For example, to find factors of 36, we check divisibility from 1 up to $\sqrt{36} = 6$.
- 1 divides 36, so 1 and 36 are factors.
- 2 divides 36, so 2 and 18 are factors.
- 3 divides 36, so 3 and 12 are factors.
- 4 divides 36, so 4 and 9 are factors.
- 5 does not divide 36.
- 6 divides 36, so 6 is a factor (paired with itself).
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. This systematic approach helps ensure all factors are found when checking the condition N = p + q + r.
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