The little buses weren’t particularly safe – no seatbelts and often not much to hold on to. On one it was so crowded that I was worried that Howard would fall out of the door as it wasn’t always closed at the point of setting off. After this we always waited for a bus that wasn’t crowded – there were so many that we knew we wouldn’t be waiting long. Bajan people are really friendly, and we often got chatting during our journey.

For this blog post my number will be 350. If this is the first of my posts you have read then the idea is to identify a number that occurs naturally in my everyday life and then find an interesting mathematical property to share. 350 is a Zumkeller number – I’d never heard of this before – and there are details below.

# Zumkeller Numbers

A Zumkeller number is one whose divisors can be partitioned into two sets with the same sum.

For example, 6 is a Zumkeller number. Its divisors are 1, 2, 3 and 6. These can be partitioned into two sets with the same sum as follows:

1. 1, 2, 3 where 1 + 2 + 3 = 6
2. 6

Another example is 20. Its divisors are 1, 2, 4, 5, 10 and 20. These can be partitioned into two groups as follows:

1. 2, 4, 5, 10 where 2 + 4 + 5 + 10 = 21
2. 20, 1 where 1 + 20 = 21

Looking at 350: its divisors are 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350. These can be partitioned into two sets with equal sum as follows:

1. 5, 7, 10, 350 where 5 + 7 + 10 + 175 = 372
2. 1, 2, 14, 25, 35, 50, 70, 175 where 1 + 2 + 14 + 25 + 35 + 50 + 70 + 175 = 372

The first Zumkeller numbers are 6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88 and so on.

All Zumkeller numbers are perfect where a perfect number is a positive integer that is equal to the sum of its proper divisors (i.e. excluding the number itself). This means that the divisors of a Zumkeller number can be partitioned into one containing the number itself and the other containing all the other divisors – such as in the case of 6 above. The reverse is also true: all perfect numbers are Zumkeller numbers.

If a Zumkeller number is not perfect, then it’s abundant: an abundant number is one where the sum of the proper divisors is greater than the number itself.

12 is a Zumkeller number and is abundant because its proper divisors are 1, 2, 3, 4, 6 and their sum is 1 + 2 + 3 + 4 + 6 = 16. These can be divided into two groups with equal sum as follows:

1. 1, 3, 4, 6 = 1 + 3 + 4 + 6 = 14
2. 2, 12 = 2 + 12 = 14

Unlike perfect numbers not all abundant numbers are Zumkeller numbers. For example, 72 is abundant but not Zumkeller. The divisors of 72 are 1, 2, 3, 4,6,8,9,12,18,24,36 and 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195 which cannot be partitioned into two. In fact, as far as I can tell, all abundant numbers that are not Zumkeller have odd proper divisor sums. Do you have any thoughts about Zumkeller numbers? Please share below.

Also, the word Zumkeller means ‘into the cellar’. Could it be applied here because it is easy to partition a Zumkeller number and store it in a cellar? I’ve no idea! How about you?