Câu hỏi của Nguyen Thi ngoc mai

Question

Tìm m;n thuộc N sao cho

2^m-2^n=2016

Nguyen Viet Dat · Accepted Answer

Vi $2^m-2^n=2016\Rightarrow2^m>2^n\Rightarrow m>n$

Dat m=n+x (x thuoc N*)ta co:

$2^m-2^n=2016$

$2^{n+k}-2^n=2016$

$2^n.2^k-2^n=2016$

$2^n\left(2^k-1\right)=2016$(1)

Vi $2^k-1$la so le $\Rightarrow2^k-1$ khong chia het cho 2 ma 2016 chia het cho 32 ma khong chia het cho 64

$\Rightarrow2^k=32$

$2^k=2^5$

$\Rightarrow k=5$

Thay k=5 vao (1) ta co:

$2^5\left(2^n-1\right)=2016$

$32\left(2^n-1\right)=2016$

$2^n-1=2016:32$

$2^n-1=63$

$2^n=63+1$

$2^n=64$

$2^n=2^6$

$\Rightarrow n=6$

Voi n=6;k=5 thi $m=6+5=11$

Vay $n=6;m=11$

Tick cho minh nha

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