How to prove the NP-completeness of 15-Partition Problem
I think this could work.

We multyply all the numbers by fifteen and we also multiply $B$, the quantity that each "bucket" should add to, by fifteen and we add twelve to it. Then we add twelve numbers of size one for each bucket to the input.

As twelve is less than fifteen, we have to use exactly twelve of them to fill each bucket, then we have to use other three numbers to fill each bucket. As random_hacker says in the comment, if we use more than twelve numbers in a bucket, we will not be able to cover the gaps in another bucket and so that bucket won´t be able to reach the quantity it should reach.

We see that if the big numbers fill the buckets in 15-partition, they also fill the buckets in 3-partition. Also, if we put four or more big numbers in a bucket, we will have more than fifteen numbers in that bucket