a)

180 = 2² x 3² x 5
= (2 x 3) x (5 x 2 x 3)
= 6 x 30

largest integer = 30

180 = 2² x 3² x 5

Prime factors 2 and 3 already have power of 2. To get the smallest perfect square we just need to make the 5 have a power of 2 as well.

Multiply 180 by 5, we would get :

180 x 5
= 2² x 3² x 5 x 5
= 2² x 3² x 5²
= (2 x 3 x 5) x (2 x 3 x 5)
= (2 x 3 x 5)²
= 30²
= 900


So smallest integer k = 5

42 = 2 x 3 x 7
So 42n = 2 x 3 x 7 x n

180 = 2² x 3² x 5
= (2 x 3) x (2 x 3 x 5)


Notice that 42 and 180 only have a (2 x 3) in common. That is basically a factor 6.

Now, the remaining prime factors are not in common.

42 has one prime factor 7 that 180 doesn't have.

Being a prime number, 7 cannot be divided exactly anymore into more smaller factors so we know there are no more common factors to take out.

180 has a (2 x 3 x 5) that 42 doesn't have. These factors also cannot be broken down into more smaller factors.


So we multiply 42 by (2 x 3 x 5) to get a multiple of 180.

Smallest n we can have is (2 x 3 x 5) = 30
such that :

42n

= 2 x 3 x 7 x 2 x 3 x 5
= 2² x 3² x 5 x 7
= 180 x 7
= 1260