Question

show that product of any two integers of the form 3k+1 is also of the same form​

Answer 1

Detailed proof of the integers is given below.

Any two integers of the form  $3k+1,k\in I$

can be written as : $3m+1,3n+1;m,n\in I$

Now we have to find the product of the two integers numbers.

$(3m+1)(3n+1)=9mn+3m+3n+1$

Now the expression can be simplified as :

$9mn+3m+3n+1=3(3mn+m+n)+1$

From the closure property of integers, we can say that the product and sum of any integer is an integer.

Hence "mn" or $(m\times n)$  is an integer, so "3mn" is also an integer.

The sum $m+n$  is also an integer, hence $3mn+m+n$ is also an integer as "mn" , "$m+n$" and 3 are all integers.

Hence we can write :

$3mn+m+n=p;p\in I$

Therefore the product of the two integers can be simplified in the form $3p+1;p\in I$ .

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