Question

# Prove that if a positive integer is of the form 6q+5, then it is of the form 3q+2 for some integer q but not conversely.

Solution

## Let n=6q+5 , where q is a positive integer. We know that any positive integer is of the form 3k , 3k+1 , 3k+2. Now , if q=3k then, n=6(3k)+5=18q+5=18q+3+2 n=3(6q+1)+2 n=3m+2 where m=6q+1 Now, if q=(3k+1) n=6(3k+1)+5 n=18q+6+5 n=18q+9+2 n=3(6q+3)+2 n=3m+2 , where m=6q+3 Now , if q=3k+2 n=6(3k+2)+5 n=18q+12+5 n=3(6q+5)+2 n=3m+2 , where m=(6q+5) Therefore , if a positive integer is of the form 6q+5 then it is of the form 3q+2. Now let n=3q+2 , where q is a positive integer. We know that any positive integer is of the form 6q , 6q+2 , 6q+3 , 6q+4 , 6q+5 Now, if q=6q n=3q+2 n=3(6q)+2 n=18q+2 n=2(9q+1) n=2m Here clearly we can observe that 3q+2 is not in the form of 6q+5. Hence we can conclude that if a positive integer is of the form 6q+5 , then it is of the form 3q+2 but not conversely.

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