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SOURCE:COMPETITION Number of Problems: 21. FOR PRINT ::: (Book)
Three positive integers are each greater than , have a product of , and are pairwise relatively prime. What is their sum?
The prime factorization of is . These three factors are pairwise relatively prime, so the sum is
For how many positive integers is a prime number?
Factoring, we get . Exactly of and must be and the other a prime number. If , then , and , which is not prime. On the other hand, if , then , and , which is a prime number. The answer is .
Suppose that is the product of three consecutive integers and that is divisible by . Which of the following is not necessarily a divisor of ?
Whenever is the product of three consecutive integers, is divisible by , meaning it is divisible by .
It also mentions that it is divisible by , so the number is definitely divisible by all the factors of .
In our answer choices, the one that is not a factor of is .
Which of the following numbers is a perfect square?
Using the fact that , we can write:
Clearly is a square, and as , , and are primes, none of the other four are squares.
What is the probability that a randomly drawn positive factor of is less than ?
For a positive number which is not a perfect square, exactly half of the positive factors will be less than .
Since is not a perfect square, half of the positive factors of will be less than .
Clearly, there are no positive factors of between and .
Therefore half of the positive factors will be less than .
So the answer is .
Testing all numbers less than , numbers , and divide . The prime factorization of is . Using the formula for the number of divisors, the total number of divisors of is . Therefore, our desired probability is