Let-a-and-b-be-positive-integers-If-118-119-5-a-b-then-what-is-a-max-

Question Number 177000 by Ar Brandon last updated on 29/Sep/22

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}. \\ $$$$\mathrm{If}\:\mathrm{118}!+\mathrm{119}!=\mathrm{5}^{{a}} {b}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:{a}_{\mathrm{max}} \:?\: \\ $$

Answered by BaliramKumar last updated on 29/Sep/22

$$\mathrm{118}!\:+\:\mathrm{119}!\:=\:\mathrm{118}!\left(\mathrm{1}+\mathrm{119}\right)\:=\:\mathrm{120}×\mathrm{118}! \\ $$$$\mathrm{5}^{\mathrm{1}} ×\mathrm{24}×\mathrm{118}!\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{|c|c|c|c|}{\mathrm{5}}&\hline{\mathrm{118}}\\{\mathrm{5}}&\hline{\mathrm{23}}\\{\mathrm{5}}&\hline{\mathrm{4}}\\{}&\hline{\mathrm{0}}\\\hline\end{array} \\ $$$${a}_{{max}} \:=\:\mathrm{1}\:+\:\mathrm{23}\:+\:\mathrm{4}\:+\:\mathrm{0}\:=\:\mathrm{28} \\ $$

Commented by Ar Brandon last updated on 29/Sep/22

Thanks! I now understand your method. I didn't get it at first.

Commented by Tawa11 last updated on 02/Oct/22

$$\mathrm{Great}\:\mathrm{sir} \\ $$

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