Question

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 1.

Answer #1

since n is odd so all factors of 3n are odd

but 16 is an even no. whose only prime factor is 2

let a be a common factor to both 3n and 3n+16

since a is one of the factor of 3n so , a divides 3n

or ,3n=ak for some integer k

also 3n+16 is divisible by a so ,

3n+16=am for some integer m

now,3(ak)+16=am

or,16=a(m-3k)

this implies 16 is divisible by a

but a is an odd number

so ,here is the contradiction

therefore 3n and 3n+16 has no common factor

i.e. gcd(3n,3n+16)=1

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