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Question Number 8359 by Nayon last updated on 09/Oct/16

prove that 10 divides 11^(10) −1

provethat10divides1110−1

Commented by Rasheed Soomro last updated on 09/Oct/16

Actually 10 divides 11^n −1 for n≥0 11^n −1=(10+1)^n −1 All the terms of the expansion of (10+1)^n contain 10 as a factor except the last term 1. So all the terms of expansion of ′(10+1)^n −1′ contain 10 as a factor. Or 10 ∣ (11^n −1).

Actually10divides11n−1forn⩾011n−1=(10+1)n−1Allthetermsoftheexpansionof(10+1)ncontain10asafactorexceptthelastterm1.Soallthetermsofexpansionof′(10+1)n−1′contain10asafactor.Or10∣(11n−1).

Answered by Rasheed Soomro last updated on 12/Oct/16

10 divides 11^(10) −1 −−−−−−−−−−−−−−−−−−−−− 11^(10) −1=(10+1)^(10) −1 = (((10)),(( 0)) )(10)^(10) + (((10)),(( 1)) )(10)^9 +...+ (((10)),(( 9)) )(10)+ (((10)),((10)) )(10)^0 −1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(10)^0 }+ (((10)),((10)) )(10)^0 −1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(10)^0 }+1−1 =(10){ (((10)),(( 0)) )(10)^9 + (((10)),(( 1)) )(10)^8 +...+ (((10)),(( 9)) )(1)} 10 is a factor of 11^(10) −1 Or 10 ∣ (11^(10) −1)

10divides1110−1−−−−−−−−−−−−−−−−−−−−−1110−1=(10+1)10−1=(100)(10)10+(101)(10)9+...+(109)(10)+(1010)(10)0−1=(10){(100)(10)9+(101)(10)8+...+(109)(10)0}+(1010)(10)0−1=(10){(100)(10)9+(101)(10)8+...+(109)(10)0}+1−1=(10){(100)(10)9+(101)(10)8+...+(109)(1)}10isafactorof1110−1Or10∣(1110−1)

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