Is 2027 prime?

Steps to Solve

1. Determine the range of possible divisors

To check if 2027 is prime, we only need to test divisibility by prime numbers up to the square root of 2027.

Calculate the square root: $$ \sqrt{2027} \approx 45.0 $$

So, we will check for divisibility using prime numbers less than or equal to 45: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43.

2. Check for divisibility by 2

2027 is an odd number, so it is not divisible by 2.

3. Check for divisibility by 3

To check for divisibility by 3, sum the digits of 2027: $$ 2 + 0 + 2 + 7 = 11 $$ Since 11 is not divisible by 3, 2027 is not divisible by 3.

4. Check for divisibility by 5

Numbers divisible by 5 end in 0 or 5. Since 2027 ends in 7, it is not divisible by 5.

5. Check for divisibility by other primes

Continue checking the other prime numbers:

• For 7: ( 2027 \div 7 \approx 289.57 ) (not divisible)
• For 11: ( 2027 \div 11 \approx 184.27 ) (not divisible)
• For 13: ( 2027 \div 13 \approx 155.92 ) (not divisible)
• For 17: ( 2027 \div 17 \approx 119.24 ) (not divisible)
• For 19: ( 2027 \div 19 \approx 106.68 ) (not divisible)
• For 23: ( 2027 \div 23 \approx 88.13 ) (not divisible)
• For 29: ( 2027 \div 29 \approx 69.90 ) (not divisible)
• For 31: ( 2027 \div 31 \approx 65.39 ) (not divisible)
• For 37: ( 2027 \div 37 \approx 54.59 ) (not divisible)
• For 41: ( 2027 \div 41 \approx 49.66 ) (not divisible)
• For 43: ( 2027 \div 43 \approx 47.11 ) (not divisible)

Since 2027 is not divisible by any of these primes, it has no divisors other than 1 and itself.

6. Conclusion

Since we found no divisors, we conclude that 2027 is a prime number.