Consider the number N = 12 6× 3 8× 5 3. Which of the following statements is/are correct? 1. The number of odd factors of N is 60. 2. The number of even factors of N is 720. Select the correct answer using the code given below :
Understanding the Problem: Finding Factors of a Number
The question asks us to determine the correctness of two statements about the factors of a given number, N. The number is expressed in a specific form: N = \(12^6 \times 3^8 \times 5^3\). We need to calculate the number of odd factors and the number of even factors of N and compare them with the values given in the statements.
Step-by-Step Solution: Calculating Factors of N
Prime Factorization of N
First, we need to express the number N in its prime factorization form. This means breaking down the base of each term into its prime components.
N = \(12^6 \times 3^8 \times 5^3\)
We know that \(12 = 4 \times 3 = 2^2 \times 3\). Substituting this into the expression for N:
N = \((2^2 \times 3)^6 \times 3^8 \times 5^3\)
Using the exponent rule \((a \times b)^m = a^m \times b^m\) and \((a^m)^n = a^{m \times n}\):
N = \((2^2)^6 \times 3^6 \times 3^8 \times 5^3\)
N = \(2^{12} \times 3^6 \times 3^8 \times 5^3\)
Using the exponent rule \(a^m \times a^n = a^{m+n}\):
N = \(2^{12} \times 3^{(6+8)} \times 5^3\)
N = \(2^{12} \times 3^{14} \times 5^3\)
So, the prime factorization of N is \(2^{12} \times 3^{14} \times 5^3\). A factor of N will be of the form \(2^a \times 3^b \times 5^c\), where \(0 \le a \le 12\), \(0 \le b \le 14\), and \(0 \le c \le 3\).
Calculating the Number of Odd Factors
An odd factor is a factor that is not divisible by 2. In the prime factorization of a number, an odd factor will not have any power of 2 in its prime factorization. Therefore, to find the number of odd factors of N = \(2^{12} \times 3^{14} \times 5^3\), we must set the exponent of 2 to 0.
An odd factor of N is of the form \(2^a \times 3^b \times 5^c\), where the power of 2 is \(a=0\). The possible values for the exponents 'b' and 'c' are:
- For the prime factor 3, the exponent 'b' can take any value from 0 to 14 (i.e., 0, 1, 2, ..., 14). The number of choices for 'b' is \(14+1 = 15\).
- For the prime factor 5, the exponent 'c' can take any value from 0 to 3 (i.e., 0, 1, 2, 3). The number of choices for 'c' is \(3+1 = 4\).
The number of odd factors is the product of the number of choices for the exponents of the odd prime factors:
Number of odd factors = (Number of choices for 'b') \(\times\) (Number of choices for 'c')
Number of odd factors = \(15 \times 4 = 60\).
Statement 1 says the number of odd factors of N is 60. Our calculation shows this is correct.
Calculating the Number of Even Factors
An even factor is a factor that is divisible by 2. In the prime factorization of a number, an even factor must have at least one power of 2. For N = \(2^{12} \times 3^{14} \times 5^3\), an even factor is of the form \(2^a \times 3^b \times 5^c\), where the power of 2, 'a', must be at least 1.
The possible values for the exponents 'a', 'b', and 'c' are:
- For the prime factor 2, the exponent 'a' can take any value from 1 to 12 (i.e., 1, 2, ..., 12). The number of choices for 'a' is \(12 - 1 + 1 = 12\). (We exclude \(2^0\) because that would result in an odd factor).
- For the prime factor 3, the exponent 'b' can take any value from 0 to 14 (i.e., 0, 1, 2, ..., 14). The number of choices for 'b' is \(14+1 = 15\).
- For the prime factor 5, the exponent 'c' can take any value from 0 to 3 (i.e., 0, 1, 2, 3). The number of choices for 'c' is \(3+1 = 4\).
The number of even factors is the product of the number of choices for each exponent:
Number of even factors = (Number of choices for 'a') \(\times\) (Number of choices for 'b') \(\times\) (Number of choices for 'c')
Number of even factors = \(12 \times 15 \times 4 = 12 \times 60 = 720\).
Statement 2 says the number of even factors of N is 720. Our calculation shows this is correct.
Alternatively, the total number of factors of N is \((12+1)(14+1)(3+1) = 13 \times 15 \times 4 = 13 \times 60 = 780\). The total number of factors is the sum of odd factors and even factors. So, Number of even factors = Total factors - Number of odd factors = \(780 - 60 = 720\). This confirms our previous calculation.
Verification of Statements
Statement 1: The number of odd factors of N is 60. Based on our calculation, this statement is correct.
Statement 2: The number of even factors of N is 720. Based on our calculation, this statement is correct.
Since both statements are correct, the correct option is "Both 1 and 2".
Revision Table: Understanding Factors
| Concept | Description | How to Calculate for \(p_1^{a_1} p_2^{a_2} \dots\) |
|---|---|---|
| Prime Factorization | Expressing a number as a product of its prime numbers raised to certain powers. | Break down the number until all bases are prime. |
| Total Number of Factors | The count of all positive integers that divide the number evenly. | \((a_1+1)(a_2+1)\dots(a_k+1)\) |
| Number of Odd Factors | The count of factors that are not divisible by 2. | Ignore the power of 2. Calculate factors from only odd prime factors: \((a_2+1)(a_3+1)\dots\) (assuming \(p_1=2\)) |
| Number of Even Factors | The count of factors that are divisible by 2. | Total Factors - Number of Odd Factors, OR (power of 2) \(\times\) (factors from odd primes) i.e., \(a_1 \times (a_2+1)(a_3+1)\dots\) |
Additional Information: Properties of Factors
Factors play an important role in number theory. Understanding how to find the number of factors, including classifying them as odd or even, is a common type of problem in competitive exams. The method relies entirely on the prime factorization of the number. If a number has no prime factor of 2 (i.e., it's an odd number), then all its factors are odd. An even number will always have both odd and even factors (unless it's a power of 2, in which case it only has even factors except for 1, which is odd - but usually, problems consider factors > 1 in certain contexts, though here total factors are considered). The number of factors formula is a direct result of how combinations of prime powers form unique factors.
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