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What is the greatest possible length of a positive integer less that 1000?
The greatest possible 'length' comes from the number with the greatest number of prime factors. The greatest number of factors is created by using the smallest prime number, 2, as a factor as many times as possible. Since 2^9=512 and 2^10=1024, the greatest possible 'length' of a positive integer less than 1000 is 9.
How many numbers which are less than 1000 have an odd number of factors?
Perfect squares have an odd number of factors. There are 31 perfect squares less than 1000.
What number less than 20 an odd number and has 2 different prime factors?
Well, darling, the number you're looking for is 15. It's less than 20, odd, and can be broken down into the prime factors 3 and 5. So, there you have it, a prime factor party with the number 15 strutting its stuff.
What are all the numbers less than 100 with exactly 3 factors that are prime?
By definition, a prime number has exactly two factors. So, there are no prime numbers with exactly three factors.
What are all the even composite numbers less than 1000?
Literally every single even number less than 1000 is a composite number other than 2 which is prime.
