Is 21 a Prime Number?
Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors. The answer is clear: "21 is not a prime number." As a product of two primes and a Fibonacci number, 21 holds special significance in mathematics, games, and cultural references. Let's explore why.
Quick Facts About 21
- Is 21 a prime number? - No
- Is 21 a composite number? - Yes
- Is 21 an odd number? - Yes
- Is 21 a perfect square? - No
- Is 21 a perfect cube? - No
- Is 21 a Fibonacci number? - Yes (8th position)
- Prime Factors of 21 - 3, 7
- Factors of 21 - 1, 3, 7, 21
Why Is 21 a Composite Number?
1. It Has More Than Two Divisors
A prime number must have only two distinct positive divisors (1 and itself). 21 has four divisors:
- Divisors of 21: 1, 3, 7, 21 → Composite
2. It's a Product of Two Primes
21 can be expressed as:
- 21 = 3 × 7 (both prime numbers)
- 21 = 1 × 21
Mathematical Significance of 21
- Fibonacci Number: 8th in sequence (0,1,1,2,3,5,8,13,21,...)
- Triangular Number: Sum of first 6 natural numbers (1+2+3+4+5+6)
- Blum Integer: Product of two distinct primes (3×7) both ≡3 mod 4
- Harshad Number: Divisible by sum of digits (2+1=3; 21÷3=7)
- Smallest Number with Exactly 8 Representations as Sum of Primes
Fun Facts About the Number 21
- In Games:
- Blackjack is nicknamed "21" (target score)
- Standard points needed to win a badminton match
- In Culture:
- 21st birthday marks legal adulthood in many countries
- "21 Guns" is a famous song by Green Day
- In Science:
- Scandium's atomic number is 21
- Human has 21 amino acids (including selenocysteine)
- In Mathematics:
- Smallest number of squares needed to tile a 7×7 grid
- Number of dots in standard die (1+2+3+4+5+6)
- In History:
- 21-gun salute is the highest honor in military tradition
- 21st Amendment repealed U.S. Prohibition
- In Nature:
- Some cicada species emerge every 21 years
- 21 is the minimum age for full brain development in humans
Problem Statements
| Property | Answer |
|---|---|
| Is 21 a Prime Number? | No |
| Is 21 a Composite Number? | Yes |
| Is 21 an Odd Number? | Yes |
| Square Root of 21 | ≈4.5826 (Irrational) |
| Square of 21 | 441 |
| Cube of 21 | 9261 |
| Binary Representation | 10101 |
| Hexadecimal Representation | 15 |
| Sum of Digits | 3 |
| Prime Factorization | 3 × 7 |
| Fibonacci Position | 8th |
Conclusion
21 is a culturally significant composite number with fascinating mathematical properties. Its appearance in the Fibonacci sequence, as a triangular number, and in game theory makes it particularly interesting to mathematicians. From blackjack tables to military honors, the number 21 permeates various aspects of human culture while maintaining its mathematical elegance as the product of two primes (3 and 7).