The number of divisors of the number 38808, exclusive of the divisors 1 and itself, is
70
Finding the Number of Divisors of 38808
The question asks for the number of divisors of the number 38808, excluding the number 1 and 38808 itself. To find this, we first need to determine the total number of divisors of 38808. The total number of divisors can be found by first getting the prime factorization of the number.
Prime Factorization of 38808
We will break down 38808 into its prime factors step by step:
- Start by dividing by the smallest prime number, 2, as 38808 is even.
- \(38808 \div 2 = 19404\)
- \(19404 \div 2 = 9702\)
- \(9702 \div 2 = 4851\)
- Now, 4851 is not divisible by 2. Check for divisibility by 3. The sum of the digits is \(4+8+5+1 = 18\), which is divisible by 3, so 4851 is divisible by 3.
- \(4851 \div 3 = 1617\)
- The sum of the digits of 1617 is \(1+6+1+7 = 15\), which is divisible by 3, so 1617 is divisible by 3.
- \(1617 \div 3 = 539\)
- 539 is not divisible by 3 (sum of digits \(5+3+9 = 17\)). It's not divisible by 5. Check for divisibility by 7.
- \(539 \div 7 = 77\)
- 77 is divisible by 7.
- \(77 \div 7 = 11\)
- 11 is a prime number.
So, the prime factorization of 38808 is \(2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7 \times 11\), which can be written in exponential form as \(2^3 \times 3^2 \times 7^2 \times 11^1\).
The prime factorization is:
\[38808 = 2^3 \times 3^2 \times 7^2 \times 11^1\]Calculating the Total Number of Divisors
If a number is expressed in its prime factorization form as \(p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}\), the total number of its positive divisors is given by the product of one more than each exponent in the prime factorization.
For 38808, the exponents are 3 (for prime 2), 2 (for prime 3), 2 (for prime 7), and 1 (for prime 11).
Total number of divisors = \((a_1+1)(a_2+1)(a_3+1)(a_4+1)\)
Total number of divisors = \((3+1)(2+1)(2+1)(1+1)\)
Total number of divisors = \(4 \times 3 \times 3 \times 2\)
Total number of divisors = \(12 \times 6\)
Total number of divisors = \(72\)
So, there are 72 positive divisors of 38808.
Finding Divisors Excluding 1 and 38808
The question asks for the number of divisors exclusive of 1 and the number itself (38808). These are the two smallest and largest divisors, respectively.
Number of required divisors = (Total number of divisors) - 2
Number of required divisors = \(72 - 2\)
Number of required divisors = \(70\)
Thus, the number of divisors of 38808, exclusive of 1 and 38808, is 70.
Summary of Steps to Find Number of Divisors
- Perform the prime factorization of the given number.
- Write the number in the form \(p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}\).
- Calculate the total number of divisors using the formula \((a_1+1)(a_2+1)\dots(a_k+1)\).
- If excluding 1 and the number itself, subtract 2 from the total.
| Step | Description | Calculation |
|---|---|---|
| 1 | Prime Factorization | \(38808 = 2^3 \times 3^2 \times 7^2 \times 11^1\) |
| 2 | Total Divisors Formula | \((3+1)(2+1)(2+1)(1+1)\) |
| 3 | Total Divisors Value | \(4 \times 3 \times 3 \times 2 = 72\) |
| 4 | Exclude 1 and 38808 | \(72 - 2 = 70\) |
The final count of divisors, excluding 1 and 38808, is 70.
Revision Table: Key Concepts for Number of Divisors
| Concept | Explanation | Example |
|---|---|---|
| Divisor | A number that divides another number exactly, leaving no remainder. | Divisors of 12 are 1, 2, 3, 4, 6, 12. |
| Prime Factorization | Expressing a number as a product of its prime factors. | \(12 = 2^2 \times 3^1\) |
| Formula for Total Divisors | If \(N = p_1^{a_1} \times \dots \times p_k^{a_k}\), total divisors \(= (a_1+1)\dots(a_k+1)\). | For \(12 = 2^2 \times 3^1\), total divisors \(= (2+1)(1+1) = 3 \times 2 = 6\). |
| Proper Divisors | Divisors of a number, excluding the number itself. | Proper divisors of 12 are 1, 2, 3, 4, 6. (Total - 1) |
| Divisors Exclusive of 1 and itself | Divisors of a number, excluding 1 and the number itself. | Divisors of 12 exclusive of 1 and 12 are 2, 3, 4, 6. (Total - 2) |
Additional Information on Finding Number of Divisors
Understanding prime factorization is fundamental to many number theory problems, including finding the number of divisors. Every positive integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
When we write a number \(N\) as \(N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}\), where \(p_1, p_2, \dots, p_k\) are distinct prime numbers and \(a_1, a_2, \dots, a_k\) are positive integers, any divisor of \(N\) must be of the form \(p_1^{b_1} \times p_2^{b_2} \times \dots \times p_k^{b_k}\), where \(0 \le b_i \le a_i\) for each \(i\).
The number of choices for each exponent \(b_i\) is \((a_i + 1)\), as \(b_i\) can take any integer value from 0 up to \(a_i\). Since the choices for each exponent are independent, the total number of combinations of these exponents is the product of the number of choices for each exponent, which is \((a_1+1)(a_2+1)\dots(a_k+1)\).
The divisor 1 corresponds to all exponents being 0 (\(p_1^0 p_2^0 \dots p_k^0 = 1\)), and the divisor \(N\) itself corresponds to all exponents being \(a_i\) (\(p_1^{a_1} p_2^{a_2} \dots p_k^{a_k} = N\)). When we exclude these two, we subtract 2 from the total count.
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