Modest Numbers (Continued)

CONTINUED FROM PREVIOUS POST

I'm having to run the previous blog post into this new post because I found that the word wrap wasn't working despite numerous efforts to fix it. 

In this continuation, I want to mention the fact that some numbers are "modest" in two ways not just one. Here is a list of such numbers in the range up to 40,000:

1333, 1999, 2333, 2666, 2999, 3999, 4666, 4999, 5999, 6999, 7999, 8999, 11111, 13333, 19999, 21111, 22222, 23333, 26666, 29999, 31111, 33333, 39999

As can be seen, all numbers have many repeated digits. Let's look at the first number in the list, 1333. We see that:$$ 1333 \! \!\! \mod 33 \equiv 13\\1333 \! \! \! \mod 333 \equiv 1$$Once we extend the range to one million, we find some numbers that are "modest" in three ways. These are:

133333, 199999, 233333, 266666, 299999, 399999, 466666, 499999, 599999, 699999, 799999, 899999

Taking the first number in the list above, 133333, we find that:$$ 133333 \! \!\! \mod 333 \equiv 133\\133333 \! \! \! \mod 3333 \equiv 13\\133333 \! \! \! \mod 33333 \equiv 1$$Clearly there is a pattern here and if we were to extend the range even further we would find that there are numbers that are modest in four ways and more. For example, 13333333 is "modest" is four ways:$$ 13333333 \! \!\! \mod 3333 \equiv 1333\\13333333 \! \! \! \mod 33333 \equiv 133\\13333333 \! \! \! \mod 333333 \equiv 13\\13333333 \! \! \! \mod 3333333 \equiv 1$$Once the algorithm for splitting any two digit number or larger into two parts is in place, it can be applied to other scenarios other than modest numbers. For example, consider this scenario where we define a "digestible" number for want of a better term as follows:

A number \(n\) is called digestible if its digits can be separated into two numbers \(a\) and \(b\) such that \( n\) divides evenly into \(a^b\).

In the range up to 40,000, there are 41 numbers that satisfy this criterion. They are (permalink):

128, 256, 486, 648, 729, 1024, 1296, 2048, 2187, 3072, 4096, 6075, 6144, 6561, 6912, 8192, 10240, 12288, 13824, 14336, 15488, 15625, 16384, 16807, 17496, 18432, 20480, 21609, 22528, 24576, 26624, 27648, 28672, 30375, 30720, 32768, 33614, 34816, 35721, 36864, 38912

The details for each number are as follows:

Figure 1

Let's take the first number in this list, 128, that divides evenly into 12 raised to the 8th power. Now 12 raised to the 8th power is 429981696 and 128 | 429981696 = 3359232. This is just an example of the sorts of investigations that can be carried out. Notice how all the powers of 2 are represented.