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Prove: For all positive integers n, the numbers 7n+ 5 and 7n+ 12 are relatively prime.

Prove: For all positive integers n, the numbers 7n+ 5 and 7n+ 12 are relatively prime.

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Answer #1

bezout's theorem  :- two integer a and b are coprime if their exist integer u and v such that au+bv=1

let b = 7n+5 , a = 7n+12

(7n+12)=(7n+5)+7

(7n+5)= n(7)+5

7= 1(5)+2

5=2(2)+1

we now just go back

1=5-2(2)

= 5-2(7-1(5))

= 5-2(7)+2(5)

= -2(7)+3(5)

= -2(7)+3((7n+5)-n(7))

= -2(7)+3(7n+5)-3n(7)

= -(2+3n)(7)+3(7n+5)

= -(2+3n)((7n+12)-(7n+5))+3(7n+5)

= -(2+3n)(7n+12)+(2+3n)(7n+5)+3(7n+5)

= -(2+3n)(7n+12)+(5+3n)(7n+5)

therefore -(2+3n)(7n+12)+(5+3n)(7n+5)=1

u = -(2+3n) v= (5+3n)

since n is integer therefore u and v are integers by above theorem we can conclude that 7n+12 and 7n+5 are relatively prime.

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