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? Choices: A) 2 B) 3 C) 4 D) 6 E) 8
First of all, think which is the smallest Mersenne prime?   The smallest prime 1 less than a power of 2 is 3.   We need the last digit of 2^{43112609} - 1. What will be the last digit of 2^{43112609}?   Recall cyclicity of last digits. After every 4 powers, the last digit repeats itself. Let's look at the last digits of powers of 2.   2^1 = 2   2^2 = 4   2^3 = 8   2^4 = 6   2^5 = 2   2^6 = 4   and so on...   When 43112609 is divided by 4, you get remainder 1 (divide just the last two digits by 4. Whatever remainder you get will be the remainder when you divide the actual number by 4)   So the last digit of 2^{43112609} is 2   Last digit of 2^{43112609} - 1 is 1   Product of the last digits is 1*3= 
<gadget id="calculator">1*3</gadget>
<output>3</output>
3   Answer (3)
<result>B</result>
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?