Bunuel wrote:
Is integer k a prime number?
(1) k = 10! + m, where 1 < m < 8
(2) k is a multiple of 7
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:On a problem such as this, you must use conceptual understanding of the number line and divisibility to answer the question. Algebraic manipulation will not help you. Number picking will not help you. And certainly just doing it will not help you; calculating the number would be impossible!
The goal from each statement is to prove whether k is prime or importantly not prime. It is impossible without computer or calculator assistance to determine whether large number is actually prime (there are too many numbers you would have to check for divisibility), but it is actually quite easy to prove that a number is not prime. If you can prove that k is divisible by anything other than 1 and itself, you have proven that k is not prime.
It is on this type of problem that you should be looking to disprove the question and find a no answer.
In statement (1), you learn that k = 10! + either 2, 3, 4, 5, 6, or 7. To review, 10! (10 factorial) represents the product of all positive numbers from 1 to 10, inclusive: 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. As you can see, the number 10! is a multiple of each of 2, 3, 4, 5, 6, and 7. For demonstration, say that m were 7. 10! + 7 will then definitely be divisible by 7. If you find any multiple of 7 on the number line and add another 7 to it, it will always still be divisible by 7. Take the number 63, a multiple of 7. If you add 7 to 63, you get 70, another multiple of 7. This is then true for any potential value of m. If m were 2, then 10! (an even number) + 2 will remain divisible by 2. If m were 3, then 10! (a multiple of 3) + 3 will remain divisible by 3. Statement (1) thus proves that k is not prime, as whatever value of m (2 through 7) we add to 10!, k will remain divisible by that number, so it cannot be a prime number. Statement (1) is sufficient.
Be careful with statement (2). If you carry some information with you from statement (1)—the fact that k is a large multiple of 7—you might think that statement (2) is also sufficient. In other words, you might think, if k is a multiple of 7, then it could never be prime, as it will be divisible by 7. However, remember that there is one multiple of 7 that is prime: 7 itself. Statement (2) is not sufficient, because k could be prime (7) or it could be any of the infinite set of multiples of 7 that are not prime.
The answer to this question is answer choice A.