Prime Numbers Identification

The question asks for the count of prime numbers between 488 and 494, inclusive. A prime number is a natural number greater than 1 that has only two distinct positive divisors: 1 and itself.

We need to check each integer in the range [488, 494]: 488, 489, 490, 491, 492, 493, 494.

Checking Numbers for Primality

• 488: Even number greater than 2, divisible by 2. Not prime.
• 489: Sum of digits = 4 + 8 + 9 = 21. Divisible by 3 (since 21 is divisible by 3). Not prime.
• 490: Ends in 0, divisible by 10 (and 2, 5). Not prime.
• 491:
  • Not divisible by 2 (odd).
  • Not divisible by 3 (sum of digits 14).
  • Not divisible by 5 (does not end in 0 or 5).
  • Check primes up to $\sqrt{491} \approx 22.16$. Primes to check: 7, 11, 13, 17, 19.
  • $491 \div 7 = 70$ remainder 1.
  • $491 \div 11 = 44$ remainder 7.
  • $491 \div 13 = 37$ remainder 10.
  • $491 \div 17 = 28$ remainder 15.
  • $491 \div 19 = 25$ remainder 16.
  • Since 491 is not divisible by any prime number less than or equal to its square root, 491 is prime.
• 492: Even number greater than 2, divisible by 2. Not prime.
• 493:
  • Not divisible by 2, 3, 5.
  • Check primes: $493 \div 17 = 29$. Divisible by 17. Not prime.
• 494: Even number greater than 2, divisible by 2. Not prime.

Final Count

The only prime number found in the range [488, 494] is 491.

Therefore, the number of prime numbers lying between 488 and 494, both included, is 1.