VJesus12 wrote: ↑Thu May 07, 2020 12:35 pm
A Mersenne number is a positive integer that is one less than any power of 2. A Mersenne prime is a Mersenne number that also happens to be prime. The largest known prime number, \(2^{43112609} - 1,\) is a Mersenne prime. If the largest known Mersenne prime were multiplied by the smallest Mersenne prime, which of the following would represent the units digit of the product?
A. 2
B. 3
C. 4
D. 6
E. 8
[spoiler]OA=B[/spoiler]
Source: Veritas Prep
Solution:
To find the units digit of the product of the largest known Mersenne prime and the smallest Mersenne prime, we just need to know the units digit of each of these two Mersenne primes. So let’s determine the smallest Mersenne prime. Since 2^1 - 1 = 1 is not a prime and 2^2 - 1 = 3 is a prime, the smallest Mersenne prime is 3, which has a units digit of 3. Now, let’s determine the units digit of 2^43112609 - 1, i.e., the largest known Mersenne prime
Recall that the units digit pattern of powers of 2 is 2-4-8-6. Since 43112609 has a remainder of 1 when it’s divided by 4 (recall that we can just use the last two digits of the a number to divide by 4 if we want to find the remainder of that number is divided by 4), the units digit of 2^43112609 is 2 and hence the units digit of 2^43112609 - 1 is 2 - 1 = 1.
Therefore, the product of the largest known Mersenne prime and the smallest Mersenne prime has a units digit of 1 x 3 = 3.
Alternate Solution:
Since 2^1 - 1 = 1 is not a prime and 2^2 - 1 = 3 is a prime, the smallest Mersenne prime is 3. The largest Mersenne prime is odd, since the only even prime is 2. Thus, the product of the smallest and the largest Mersenne prime is also odd. Since the product is odd, so is its units digit. The only odd option is 3.
Answer: B 
