Decide whether it is true that if n is a positive integer an

Decide whether it is true that if n is a positive integer and d is a divisor of then there is an integer a with |a| = d. Give reasons for your answer.

Solution

If n is a positive integer and d is a divisor of n, then we have n = bd for some integer b. Also, if d is positive, then b also will be positive as n is positive. Further, if n = bd, then d = n/b = a ( say) . Now, since d is a divisor of n, b will also be a divisor of n and therefore, a = n/b is an integer. Now, it is apparent that |a| = d.