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let s(n)=32^(2n-1) +1 when n=1 32^(2-1) +1 = 33 = 11*3 which is divisible by 11 i.e. s(1) is true assume s(k) is true i.e. 32^(2k-1) + 1 =11m when n=k+1 32^[2(k+1)-1] + 1 =32^(2k-1+2) + 1 =32^2 * 32^(2k-1) + 1 =32^2 *[32^(2k-1) +1] -32^2 +1 =32^2 *(11m) -1023 =32^2 *(11m) - 11*93 =11*(32^2 m -93) ∵ m is an integer ∴32^2 m -93 is an integer i.e. s(k+1) is divisible by 11 i.e. s(k+1) is true By MI, s(n) is true for all postive integer n 2. 2^995 +1 =2^(10 *100 -5) +1 =2^{5[2(100)-1]} +1 =32^[2(100)-1] +1 i.e. 2^995 +1 is divisible by 11
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1.(32^2n-1 )+1 is divisible by 11 2. (2^995)+1 is divisible by 11 請列步驟最佳解答:
let s(n)=32^(2n-1) +1 when n=1 32^(2-1) +1 = 33 = 11*3 which is divisible by 11 i.e. s(1) is true assume s(k) is true i.e. 32^(2k-1) + 1 =11m when n=k+1 32^[2(k+1)-1] + 1 =32^(2k-1+2) + 1 =32^2 * 32^(2k-1) + 1 =32^2 *[32^(2k-1) +1] -32^2 +1 =32^2 *(11m) -1023 =32^2 *(11m) - 11*93 =11*(32^2 m -93) ∵ m is an integer ∴32^2 m -93 is an integer i.e. s(k+1) is divisible by 11 i.e. s(k+1) is true By MI, s(n) is true for all postive integer n 2. 2^995 +1 =2^(10 *100 -5) +1 =2^{5[2(100)-1]} +1 =32^[2(100)-1] +1 i.e. 2^995 +1 is divisible by 11
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