`a)`
`13 . 12 + 26 . 27`
`= 13 . 12 + 13 . 2 . 27`
`= 13 . (12 + 2 . 27)`
`= 13 . (12 + 54)`
`= 13 . 66`
`= 13 . 2 . 66 \vdots 13 . 33`
`=> 13 . 12 + 26 . 27 \vdots 13 . 33` `(đpcm)`
`b)`
`6^5 . 5 - 3^5`
`= 2^5 . 3^5 . 5 - 3^5 . 1`
`= 3^5 . (2^5 . 5 - 1)`
`= 3^5 . (32 . 5 - 1)`
`= 3^5 . (160 - 1)`
`= 3^5 . 159`
`= 3^5 . 53 . 3 \vdots 53`
`=> 6^5 . 5 - 3^5 \vdots 53` `(đpcm)`
`c)`
`@` `TH_{1} :`
`2 + 2^2 + 2^3 + 2^4 + ... + 2^120`
`= (2 + 2^2) + (2^3 + 2^4) + ... + (2^119 + 2^120)`
`= 3 . 2 + 3 . 2^2 + ... + 3 . 2^119`
`= 3 . (2 + 2^2 + 2^119) \vdots 3`
`=> 2 + 2^2 + 2^3 + 2^4 + ... + 2^120 \vdots 3` `(đpcm)`
`@` `TH_{2} :`
`2 + 2^2 + 2^3 + 2^4 + ... + 2^120`
`= (2 + 2^2 + 2^3) + (2^4 + 2^5 + 2^6) + ... + (2^118 + 2^119 + 2^120)`
`= 2 . (1 + 2 + 2^2) + 2^4 . (1 + 2 + 2^2) + ... + 2^118 . (1 + 2 + 2^2)`
`= 2 . 7 + 2^4 . 7 + ... + 2^118 . 7`
`= (2 + 2^4 + ... + 2^118) . 7 \vdots 7`
`=> 2 + 2^2 + 2^3 + 2^4 + ... + 2^120 \vdots 7` `(đpcm)`
`@` `TH_{3} :`
`2 + 2^2 + 2^3 + 2^4 + ... + 2^120`
`= (2 + 2^2 + 2^3 + 2^4 + 2^5) + ... + (2^116 + 2^117 + 2^118 + 2^119 + 2^120)`
`= 2 . (1 + 2^2 + 2^3 + 2^4) + ... + 2^116 . (1 + 2^2 + 2^3 + 2^4)`
`= 2 . 31 + .. + 2^116 . 31`
`= (2 + ... + 2^11) . 32 \vdots 31`
`=> 2 + 2^2 + 2^3 + 2^4 + ... + 2^120 \vdots 31` `(đpcm)`
`@` `TH_{4} :`
`2 + 2^2 + 2^3 + 2^4 + ... + 2^120`
`= (2 + 2^5) + (2^2 + 2^6) + ... + (2^116 + 2^120)`
`= 2 . (1 + 16) + 2^2 . (1 + 16) + ... + 2^116 . (1 + 16)`
`= 2 . 17 + 2^2 . 17 + ... + 2^116 . 17`
`= 2 . (2 + 2^2 + ... + 2^116) \vdots 17`
`=> 2 + 2^2 + 2^3 + 2^4 + ... + 2^120 \vdots 17` `(đpcm)`
