MathDB

We are given

n

identical cubes, each of size

1\times 1\times 1

. We arrange all of these

n

cubes to produce one or more congruent rectangular solids, and let

B(n)

be the number of ways to do this.
For example, if

n=12

, then one arrangement is twelve

1\times1\times1

cubes, another is one

3\times 2\times2

solid, another is three

2\times 2\times1

solids, another is three

4\times1\times1

solids, etc. We do not consider, say,

2\times2\times1

and

1\times2\times2

to be different; these solids are congruent. You may wish to verify, for example, that

B(12) =11

.
Find, with proof, the integer

m

such that

10^m<B(2015^{100})<10^{m+1}

Problem Statement