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Question-108430




Question Number 108430 by Rasikh last updated on 16/Aug/20
Answered by Sarah85 last updated on 17/Aug/20
42! has got more 2s than 3s, so we have to  count the 3s  3, 6, 12, 15, 21, 24, 30, 33, 39, 42 / 10 3s  9, 18, 36 / 6 3s  27 / 3 3s  ⇒ n=19
$$\mathrm{42}!\:\mathrm{has}\:\mathrm{got}\:\mathrm{more}\:\mathrm{2s}\:\mathrm{than}\:\mathrm{3s},\:\mathrm{so}\:\mathrm{we}\:\mathrm{have}\:\mathrm{to} \\ $$$$\mathrm{count}\:\mathrm{the}\:\mathrm{3s} \\ $$$$\mathrm{3},\:\mathrm{6},\:\mathrm{12},\:\mathrm{15},\:\mathrm{21},\:\mathrm{24},\:\mathrm{30},\:\mathrm{33},\:\mathrm{39},\:\mathrm{42}\:/\:\mathrm{10}\:\mathrm{3s} \\ $$$$\mathrm{9},\:\mathrm{18},\:\mathrm{36}\:/\:\mathrm{6}\:\mathrm{3s} \\ $$$$\mathrm{27}\:/\:\mathrm{3}\:\mathrm{3s} \\ $$$$\Rightarrow\:{n}=\mathrm{19} \\ $$
Commented by Rasikh last updated on 17/Aug/20
thans a lot sir
$$\mathrm{thans}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{sir} \\ $$
Commented by udaythool last updated on 17/Aug/20
There are 14 no.s which are  multiple of 3, 4 multiple of 9 and  1 multiple of 27.  Therefore 42! is a multiple of 3^(14+4+1) =3^(19) .  No doubt 42! is a multiple of  more than 19th power of 2. Thus  42! is a multiple of  19th power of 6, since 2 and 3  are relatively prime.
$$\mathrm{There}\:\mathrm{are}\:\mathrm{14}\:\mathrm{no}.\mathrm{s}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{multiple}\:\mathrm{of}\:\mathrm{3},\:\mathrm{4}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{9}\:\mathrm{and} \\ $$$$\mathrm{1}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{27}. \\ $$$$\mathrm{Therefore}\:\mathrm{42}!\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3}^{\mathrm{14}+\mathrm{4}+\mathrm{1}} =\mathrm{3}^{\mathrm{19}} . \\ $$$$\mathrm{No}\:\mathrm{doubt}\:\mathrm{42}!\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{more}\:\mathrm{than}\:\mathrm{19th}\:\mathrm{power}\:\mathrm{of}\:\mathrm{2}.\:\mathrm{Thus} \\ $$$$\mathrm{42}!\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{19th}\:\mathrm{power}\:\mathrm{of}\:\mathrm{6},\:\mathrm{since}\:\mathrm{2}\:\mathrm{and}\:\mathrm{3} \\ $$$$\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$
Commented by Rasikh last updated on 18/Aug/20
thank you
$$\mathrm{thank}\:\mathrm{you} \\ $$

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