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Question Number 213261 by lmcp1203 last updated on 02/Nov/24
if between  10^4  and 10^n  there are 9999000 coprime numbers with 20. find n.  please. thanks
ifbetween104and10nthereare9999000coprimenumberswith20.findn.please.thanks
Answered by MrGaster last updated on 02/Nov/24
10^4 <x<10^n ,(x,20)=1  20=2^2 ×5  φ(20)=20×(1−(1/2))×(1−(1/5))=8  (((10^n −10^4 ))/(20))×φ(20)=9999000  (((10^n −10^4 ))/(20))×8=9999000  (10^n −10^4 )=((9999000×20)/8)  10^n −10^4 =24997500  10^n =25000000+10^4   10^n =25010000  n=log_(10) (2.501×10^7 )  n=log_(10) (2.501)+log_(10) (10^7 )  n=log_(10) (2.501)+7  n≈0.398+7  n≈7.398  since n must be an integer,                     determinant (((n=8)))
104<x<10n,(x,20)=120=22×5ϕ(20)=20×(112)×(115)=8(10n104)20×ϕ(20)=9999000(10n104)20×8=9999000(10n104)=9999000×20810n104=2499750010n=25000000+10410n=25010000n=log10(2.501×107)n=log10(2.501)+log10(107)n=log10(2.501)+7n0.398+7n7.398sincenmustbeaninteger,n=8
Commented by mr W last updated on 02/Nov/24
but between 10^4  and 10^8  there are  39 996 000 coprimes with 20, not  9 999 000.
butbetween104and108thereare39996000coprimeswith20,not9999000.
Answered by lmcp1203 last updated on 02/Nov/24
thank uou.
thankuou.
Answered by mr W last updated on 02/Nov/24
20=2^2 ×5  a coprime with 20 should not contain  the prime factor 2 and 5. that means  all numbers between 10^4  and 10^n ,  which are a multiple of 2 or of 5 or  of both must be eliminated.  (10^n −10^4 )(1−(1/2)−(1/5)+(1/(10)))=9999000  10^n =25 007 500  this is not possible, since n is natural  number.  that means it is not possible that  there are exactly 9999000 coprimes  with 20 between 10^4  and 10^n !
20=22×5acoprimewith20shouldnotcontaintheprimefactor2and5.thatmeansallnumbersbetween104and10n,whichareamultipleof2orof5orofbothmustbeeliminated.(10n104)(11215+110)=999900010n=25007500thisisnotpossible,sincenisnaturalnumber.thatmeansitisnotpossiblethatthereareexactly9999000coprimeswith20between104and10n!

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