j_shreyans wrote:A "Sophie Germain" prime is any positive prime number p for which 2p+1 is also prime. The product of all the possible units digits of Sophie Germain primes greater than 5 is
A)3
B)7
C)21
D)27
E)189
Prime numbers greater than 5:
7, 11, 13, 17, 19, 23, 29, 31, 37...
In each case, the units digit is either 1, 3, 7, or 9.
If p=11, then 2p + 1 = 23, which is prime.
Thus, 11 is a Sophie Germain prime.
If p=13, then 2p + 1 = 27, which is NOT prime.
If p=23, then 2p + 1 = 47, which is prime.
Thus, 23 is a Sophie Germain prime.
If p=7, then 2p + 1 = 15, which is NOT prime.
If p=17, then 2p + 1 = 35, which is NOT prime.
Note the PATTERN:
If the units digit of p is 7, then the units digit of 2p + 1 will be 5, with the result that 2p+1 will not be prime.
Thus, it is not possible for a Sophie Germain prime to have a units digit of 7.
If p=19, then 2p + 1 = 39, which is NOT prime.
If p=29, then 2p + 1 = 59, which is prime.
Thus, 29 is a Sophie Germain prime.
The units digit of a Sophie Germain prime can be 1, 3, or 9.
The product of these options = 1*3*9 = 27.
The correct answer is
D.
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