What is the largest prime number less than 1000?

Answer 1

997.

To test that 997 is prime, we only need to test values up [997^.5], so up to 31. Using divisibility rules, we can immediately eliminate several possible divisors.

Rule for 7: If you have a number, separate the last digit from the proceeding ones. Subtract twice the last digit from the number created from slicing off the last part. if that's divisible by seven, the whole number is. For example, take 343. 34-2(3) is 28, which is divisible by 7, so 343 is divisible by 7

Rule for 3: sum of digits is divisible by 3.

Rule for 2: last digit is divisible by 2.

Rule for 2^n: last n digits form a number that is divisible by 2^n

Rule for 5: last digit is 5 or 0.

Rule for 11: Difference of alternating sums of the digits, 432113 is divisible by 11 because (4+2+1)-(3+1+3) is divisible by eleven.

Rule for 9: sum of digits is divisible by 9.

Trying 13, 17, 19, 23, and 29, we see these all fail. So 997 is indeed prime.