emmak wrote:

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

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 = 3

Answer (B)

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