gmattesttaker2 wrote:
Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, ..., x + 59}?
A) 0
B) 17
C) 18
D) 23
E) 24
As Mitch pointed out, it's very hard to prove that an extremely large integer is prime.
That said, it is relatively easy to prove that
some of the values are NOT prime (aka composite)
One approach is to use a nice rule that says:
If N is divisible by d, then (N + d) is also divisible by d
For example, since 28 is divisible by 7, we know that (28 + 7) is also divisible by 7.
Since x = (2)(3)(5)(7)(11)....(53)(59), we know that x is divisible by 2, 3, 5, 7, 11, ..., and 53, 59
So, from the
above rule, we can be certain that x + 2 is divisible by 2, x + 3 is divisible by 3, x + 5 is divisible by 5, . . . and x + 59 is divisible by 59.
At this point, we've can already conclude that all values in the form (x + prime) are composite numbers.
We can prove that other values are composite as well.
For example, since x = (2)(
3)(5)(
7)(11)....(53)(59), we know that x is divisible by
21. So, from the
above rule, we can be certain that x + 21 is divisible by 21, which means x + 21 is composite.
For example, since x = (2)(
3)(
5)(7)(11)....(53)(59), we know that x is divisible by
15. So, from the
above rule, we can be certain that x + 15 is divisible by 15, which means x + 15 is composite.
We can use the two techniques above to eliminate enough of the values to conclude (by the process of elimination) that the correct answer is
A
Cheers,
Brent
