What are distinct prime factors of the number 26381 ?
23, 31, 37
Finding the Distinct Prime Factors of 26381
To find the distinct prime factors of a number, we need to perform prime factorization. Prime factorization is the process of breaking down a composite number into a product of its prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (examples: 2, 3, 5, 7, 11, etc.).
We will start dividing 26381 by the smallest prime numbers and work our way up.
- Start with 2: 26381 is an odd number, so it is not divisible by 2.
- Try 3: The sum of the digits of 26381 is \(2+6+3+8+1 = 20\). Since 20 is not divisible by 3, 26381 is not divisible by 3.
- Try 5: The number does not end in 0 or 5, so it is not divisible by 5.
- Try 7: \(26381 \div 7 = 3768\) with a remainder of 5. Not divisible by 7.
- Try 11: \(26381 \div 11 = 2398\) with a remainder of 3. Not divisible by 11.
- Try 13: \(26381 \div 13 = 2030\) with a remainder of 11. Not divisible by 13.
- Try 17: \(26381 \div 17 = 1551\) with a remainder of 14. Not divisible by 17.
- Try 19: \(26381 \div 19 = 1388\) with a remainder of 9. Not divisible by 19.
- Try 23: Let's perform the division:
| Division Step | Result | Notes |
|---|---|---|
| \(26381 \div 23\) | 1147 | Exact division |
So, 23 is a prime factor of 26381. Now we need to find the prime factors of 1147.
- Continue dividing 1147 by prime numbers, starting from 23 (or potentially smaller primes if we weren't sure 23 was the first). Let's try primes greater than or equal to 23.
- Try 23: \(1147 \div 23 = 49\) with a remainder of 20. Not divisible by 23.
- Try 29: \(1147 \div 29 = 39\) with a remainder of 16. Not divisible by 29.
- Try 31: Let's perform the division:
| Division Step | Result | Notes |
|---|---|---|
| \(1147 \div 31\) | 37 | Exact division |
So, 31 is a prime factor of 1147 (and thus of 26381). Now we need to find the prime factors of 37.
- Try 37: 37 is a prime number itself.
| Division Step | Result | Notes |
|---|---|---|
| \(37 \div 37\) | 1 | We stop when we reach 1 |
The prime factorization of 26381 is \(23 \times 31 \times 37\).
The distinct prime factors are the unique prime numbers in this factorization.
The distinct prime factors of 26381 are 23, 31, and 37.
Revision Table: Prime Factors of 26381
| Number | Smallest Prime Divisor | Result of Division |
|---|---|---|
| 26381 | 23 | 1147 |
| 1147 | 31 | 37 |
| 37 | 37 | 1 |
Additional Information: Understanding Prime Factors
A prime factor is a prime number that divides a given number exactly. For example, the prime factors of 12 are 2 and 3, because \(12 = 2 \times 2 \times 3 = 2^2 \times 3\). The distinct prime factors of 12 are 2 and 3.
Every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers. This representation is unique, ignoring the order of the factors. This is known as the Fundamental Theorem of Arithmetic.
Finding prime factors is useful in various mathematical concepts, such as finding the greatest common divisor (GCD) or the least common multiple (LCM) of two or more numbers.
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