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A Mersenne number is a positive integer that is one less tha

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A Mersenne number is a positive integer that is one less tha [#permalink] New post 11 Mar 2013, 22:25
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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 12 Mar 2013, 03:50, edited 1 time in total.
Edited the question.
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Re: A Mersenne number is a positive integer that is one less tha [#permalink] New post 11 Mar 2013, 22:43
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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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Re: A Mersenne number is a positive integer that is one less tha [#permalink] New post 12 Mar 2013, 00:21
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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


The smallest Mersenne prime is 3. Now we know that any power of 2 ends with a even integer. Thus the units digit of (2^[43112609])-1) will be an odd integer. This when multiplied by 3 will still be an odd integer. Thus the answer is the only odd integer.

B.
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Re: A Mersenne number is a positive integer that is one less tha [#permalink] New post 12 Mar 2013, 03:55
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vinaymimani wrote:
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


The smallest Mersenne prime is 3. Now we know that any power of 2 ends with a even integer. Thus the units digit of (2^[43112609])-1) will be an odd integer. This when multiplied by 3 will still be an odd integer. Thus the answer is the only odd integer.

B.


Good solution.

The smallest Mersenne prime = 2^2 - 1 = 3 = Odd;
2^(43112609) - 1 = Even - Odd = Odd;

Odd*Odd = Odd --> the units digit must be odd. Only B fit.

Answer: B.
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Re: A Mersenne number is a positive integer that is one less tha [#permalink] New post 04 May 2013, 22:23
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


The smallest Mersenne Prime is 2^2 - 1 = 3
We know that any prime other than '2' is odd.
Hence the product of these two primes should be odd.

From the options, only option B is possible.

Hence option B is the answer. :)
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Re: A Mersenne number is a positive integer that is one less tha [#permalink] New post 30 May 2014, 05:29
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Re: A Mersenne number is a positive integer that is one less tha   [#permalink] 30 May 2014, 05:29
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