Give a direct proof.
Let x be an integer. If 11x-5 is odd, then x is even.
Asuume 11x-5 is odd
11x-5= (2k+1)
so, x= 2k+1
11(2k+1)-5
22k+1-5 = 22k-4
Since 22k-4 is an integer 11x-5 is odd
Give an indirect proof.
Assume that x is odd. Then 2k+1 for some integer k
so, 11x-5= 11(2k+1)-5 = ? I am a bit confused with this one.
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Let x be an integer. Prove that 5x-11 is even if and only if x is odd.
Direct proof: Assume x is odd, then x=2k+1, for some integer k
so, 5(2k+1)-11 = 10k+5-11 = 10k-6= 2(5k-3)
Since (5k-3) is an integer, 5x-11 is even.
Assume x is even. Then x= 2l, for some integer l . Therefore, 5(2l)-11= 10l-11=
10l-12+1= 2(5l-6)+1
Since 5l-6 is an integer, 5x-11 is odd.
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Let x be an integer. Prove that x^3 is even if and only if x is even.
Proof: Assume that x is even. Then x=2k for some integer k.
Therefore, x^3=(2k)^3 = 8k^3= 2(4k^3)
Because 4k^3 is an integer, then integer k^3 is even.
For the Converse, assume x is odd. x=(2l+1), for some integer l.
x^3= (2l+1)^3= (2l+1)(2l+1)(2l+1)= 8l^3+12l^2+6l+1= 2(4l^3+6l^2+3l)+1
since (4l^3+6l^2+3l) is an integer, x^3 is even
2006-10-01
09:54:06
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5 answers
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asked by
Shivers20
2