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4-61-4-62-4-63-4-64-is-divisible-by-1-17-2-3-3-11-4-13-Mastermind-




Question Number 168537 by Mastermind last updated on 12/Apr/22
4^(61) +4^(62) +4^(63) +4^(64 )  is divisible by  (1) 17          (2) 3  (3) 11          (4) 13    Mastermind
$$\mathrm{4}^{\mathrm{61}} +\mathrm{4}^{\mathrm{62}} +\mathrm{4}^{\mathrm{63}} +\mathrm{4}^{\mathrm{64}\:} \:{is}\:{divisible}\:{by} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{17}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\mathrm{3} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{11}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{4}\right)\:\mathrm{13} \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by MJS_new last updated on 12/Apr/22
4^(61) (1+4+4^2 +4^3 )=4^(61) ×85=2^(122) ×5×17
$$\mathrm{4}^{\mathrm{61}} \left(\mathrm{1}+\mathrm{4}+\mathrm{4}^{\mathrm{2}} +\mathrm{4}^{\mathrm{3}} \right)=\mathrm{4}^{\mathrm{61}} ×\mathrm{85}=\mathrm{2}^{\mathrm{122}} ×\mathrm{5}×\mathrm{17} \\ $$
Answered by mathfreak last updated on 12/Apr/22
4^(61) +4^(62) +4^(63) +4^(64) = 4^(61) (1+4+16+64)  = 4^(61) (85) = 5(17)(4^(61) )  answer = 17
$$\mathrm{4}^{\mathrm{61}} +\mathrm{4}^{\mathrm{62}} +\mathrm{4}^{\mathrm{63}} +\mathrm{4}^{\mathrm{64}} =\:\mathrm{4}^{\mathrm{61}} \left(\mathrm{1}+\mathrm{4}+\mathrm{16}+\mathrm{64}\right) \\ $$$$=\:\mathrm{4}^{\mathrm{61}} \left(\mathrm{85}\right)\:=\:\mathrm{5}\left(\mathrm{17}\right)\left(\mathrm{4}^{\mathrm{61}} \right) \\ $$$${answer}\:=\:\mathrm{17} \\ $$

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