Consider the following statements in respect of all factors of 360 : 1. The number of factors is 24. 2. The sum of all factors is 1170. Which of the above statements is/are correct ?
Both 1 and 2
Understanding Factors of 360
The question asks us to consider two statements about the factors of the number 360. We need to determine if these statements are correct. To do this, we first need to understand how to find the factors of a number, specifically the number of factors and the sum of its factors.
Finding the Prime Factorization of 360
To find the factors of any number, it's best to start by finding its prime factorization. Prime factorization is breaking down a number into the product of its prime numbers.
Let's find the prime factors of 360:
- \(360 = 10 \times 36\)
- \(10 = 2 \times 5\) (Both are prime)
- \(36 = 6 \times 6\)
- \(6 = 2 \times 3\) (Both are prime)
So, \(360 = (2 \times 5) \times (2 \times 3) \times (2 \times 3)\)
Combining the prime factors:
\(360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\)
In exponential form, the prime factorization is:
\(360 = 2^3 \times 3^2 \times 5^1\)
Analyzing Statement 1: Number of Factors
Statement 1 says: The number of factors is 24.
To find the number of factors of a number given its prime factorization \(N = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}\), the formula is \((a_1+1)(a_2+1)\dots(a_k+1)\).
For 360, the prime factorization is \(2^3 \times 3^2 \times 5^1\). Here, the exponents are \(a_1=3\), \(a_2=2\), and \(a_3=1\).
Using the formula, the number of factors is:
\((3+1)(2+1)(1+1) = 4 \times 3 \times 2\)
Calculation:
\(4 \times 3 = 12\)
\(12 \times 2 = 24\)
So, the number of factors of 360 is 24. Statement 1 is correct.
| Prime Factor | Exponent (\(a_i\)) | (\(a_i+1\)) |
|---|---|---|
| 2 | 3 | \(3+1=4\) |
| 3 | 2 | \(2+1=3\) |
| 5 | 1 | \(1+1=2\) |
Total number of factors = Product of (\(a_i+1\)) values = \(4 \times 3 \times 2 = 24\).
Analyzing Statement 2: Sum of Factors
Statement 2 says: The sum of all factors is 1170.
To find the sum of factors of a number given its prime factorization \(N = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}\), the formula is \((1+p_1+\dots+p_1^{a_1})(1+p_2+\dots+p_2^{a_2})\dots(1+p_k+\dots+p_k^{a_k})\).
For 360, with prime factorization \(2^3 \times 3^2 \times 5^1\), the sum of factors is:
\((1+2^1+2^2+2^3)(1+3^1+3^2)(1+5^1)\)
Let's calculate each part:
- Sum for prime 2: \(1 + 2 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15\)
- Sum for prime 3: \(1 + 3 + 3^2 = 1 + 3 + 9 = 13\)
- Sum for prime 5: \(1 + 5 = 6\)
The sum of all factors is the product of these sums:
\(15 \times 13 \times 6\)
Calculation:
\(15 \times 13 = 195\)
\(195 \times 6 = 1170\)
So, the sum of all factors of 360 is 1170. Statement 2 is correct.
| Prime Factor \(p_i\) | Exponent \(a_i\) | Sum of powers \((1+p_i+\dots+p_i^{a_i})\) |
|---|---|---|
| 2 | 3 | \(1+2+2^2+2^3 = 1+2+4+8 = 15\) |
| 3 | 2 | \(1+3+3^2 = 1+3+9 = 13\) |
| 5 | 1 | \(1+5 = 6\) |
Sum of factors = \(15 \times 13 \times 6 = 1170\).
Conclusion
Based on our calculations, both Statement 1 (The number of factors is 24) and Statement 2 (The sum of all factors is 1170) are correct for the number 360.
Revision Table: Properties of 360 Factors
| Property | Calculation Method | Result for 360 | Statement Status |
|---|---|---|---|
| Prime Factorization | Break down into prime numbers | \(2^3 \times 3^2 \times 5^1\) | N/A |
| Number of Factors | \((a_1+1)(a_2+1)\dots\) | \((3+1)(2+1)(1+1) = 24\) | Statement 1 Correct |
| Sum of Factors | \((1+p_1+\dots)(1+p_2+\dots)\dots\) | \((1+2+4+8)(1+3+9)(1+5) = 15 \times 13 \times 6 = 1170\) | Statement 2 Correct |
Additional Information: Number Theory Concepts
Understanding factors, prime factorization, number of factors, and sum of factors are key concepts in number theory. These methods can be applied to any positive integer to determine its divisors and related properties.
- Factors (or Divisors): These are numbers that divide the given number evenly without leaving a remainder. For 360, examples are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. Counting these manually also gives 24 factors.
- Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).
- Composite Number: A natural number greater than 1 that is not prime (e.g., 4, 6, 8, 9, 10). 360 is a composite number.
- Formula Derivation: The formulas for the number and sum of factors come from the fact that any factor of \(N = p_1^{a_1} \dots p_k^{a_k}\) must be of the form \(p_1^{b_1} \dots p_k^{b_k}\), where \(0 \le b_i \le a_i\) for each \(i\). The number of choices for each exponent \(b_i\) is \((a_i+1)\), leading to the product for the total number of factors. The sum formula comes from the expansion of the product \((1+p_1+\dots+p_1^{a_1})\dots(1+p_k+\dots+p_k^{a_k})\), where each term in the expanded product is a unique factor of N.
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