TrueFalse In the nert eight problems tell which statements a

True-False. In the nert eight problems, tell which statements are true and give counterezamples for those that are false. Assume a, b, c, and d are arbitrary integers with a >0 and c and d nonzero.

Solution

23.

True

Assume b is a divisor of d

Hence, b is a divisor of c and d ie b is a common divisor of b and d

But, (c,d) is greatest common divisor of c and d

and b>(c,d) and a common divisor

Hence a contradiction

HEnce, b is not a divisor of d

24.

True

[c,d] is least common multiple of c and d

b is a multiple of c and b<[c,d]

So,if b were a multiple of d then b would be a common multiple of c and d smaller than [c,d] which is not possible

Hence, b is not a multiple of d