My Math Forum The last two digits

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 August 26th, 2012, 10:26 PM #1 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 The last two digits $\text {n=2m+1,\,\, n\in N,\,\, m \,\,is \,\,a\,\, non-negative \,\,integer\,\,(m \in N_0)}$ $\text{Find \,\, the\,\, last\,\, two\,\,digits\,\, of}\,\,:$ $\text{2^{2n}(2^{2n+1}-1)}$ Ans:28
 August 27th, 2012, 09:42 AM #2 Senior Member   Joined: Jun 2011 Posts: 164 Thanks: 0 Re: The last two digits I just saw this problem, have not thought about it yet, but I will give it try right here: if we just replace n with m, we get: (2^(4m+2))(2^(4m+3)-1) and we know m is a multiple of 0.5 that is greater or equal to 0 because that is the only way n will be a natural number now if we set 4m+2=2n=x, therefore 4m+3=x+1 we have 2^x(2^(x+1)-1) =(2^x)(2^x*2-1) since n is a natural number, x must be a natural number that is even so: 2^2*(2^3-1)=28 2^4*(2^5-1)=496 2^6*(2^7-1)=8128 2^8(2^9-1)=130816 2^10(2^11-1)=...28 I have a question now, when n=2, the last two digits are 96, could you please check your problem? Or did I do something wrong?
 August 27th, 2012, 05:01 PM #3 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: The last two digits wuzhe : $n=2m+1, \,\, n\in N$ so n is an odd number, not even numbers
 August 27th, 2012, 06:15 PM #4 Senior Member   Joined: Jun 2011 Posts: 164 Thanks: 0 Re: The last two digits But I thought m could be 0.5 or 1.5, did I miss something basic?
August 27th, 2012, 06:58 PM   #5
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Re: The last two digits

Quote:
 wuzhe wrote :But I thought m could be 0.5 or 1.5, did I miss something basic?
Sorry! I did not make it clear , where m is a non-negative integer $\text{(0\leq m \in Z, \,\, or\,\,m\in N_0)}$

I have edited the original post

 August 27th, 2012, 07:02 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: The last two digits If m is to be a non-negative integer, then you could use $m\in\mathbb{N_0}$ or $0\le m\in\mathbb{Z}$. $\mathbb{Z}$ is used to denote the integers. $\mathbb{N}$ is used to denote the natural numbers. $\mathbb{N_0}$ is used to denote the natural numbers and includes zero as well, i.e., the whole numbers.
 August 27th, 2012, 07:06 PM #7 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: The last two digits MarkFL : Thanks
 August 27th, 2012, 07:30 PM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: The last two digits Another off-topic suggestion: To use $\LaTeX$ to display text, use the text command: The code \text{ your text here } produces $\text{ your text here }$ This way, your text is not italicized like variables.
 August 27th, 2012, 07:39 PM #9 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: The last two digits Hint :to prove this, one of the possible solutions you may use Mathematical Induction (MI), please try it I will post using another method later.

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