cho tổng M= 2 + 2^2 + 2^3 +...+ 2^12+ 2^13 + 2^14. hỏi M có chia hết cho 2 + 2^3 + 2^5 + 2^7 + 2^9 + 2^11 + 2^13 không vì sao

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Đáp án:

`M \vdots  2+2^3+2^5+2^7+2^9+2^11+2^13`

Giải thích các bước giải:

Ta có:

`M=2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10+2^11+2^12+2^13+2^14`

`=> 2M = 2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10+2^11+2^12+2^13+2^14 + 2^15`

`=> M = 2^15 - 2` 

Gọi `2+2^3+2^5+2^7+2^9+2^11+2^13 = A`

Ta có: 

`A = 2+2^3+2^5+2^7+2^9+2^11+2^13`

`=> 4A = 2^3+2^5+2^7+2^9+2^11+2^13 + 2^15`

`=> 3A = 2^15 - 2`

`=> A =(2^15 - 2)/3`

Ta lại có: 

`(2^15 - 2) : ((2^15 - 2)/3) = 3`

`=> M \vdots  2+2^3+2^5+2^7+2^9+2^11+2^13`

Vậy `M \vdots  2+2^3+2^5+2^7+2^9+2^11+2^13`