Explain 'expressing a number using its digits'

Question

While studying divisibilty and prime numbers in my maths book (IB Mathematic Higher Level Option 10: Discrete Mathematics), I came across an explanation of a way to '[express] a number using its digits'.

It says:

If $N$ is a $k$-digit number with digits $a_k, a_{k-1}...a_2, a_1, a_0$, then
$$N = 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10^2 a_2 + 10 a_1 + a_0.$$
Shorthand $N = (a_k a_{k-1} \cdots a_2 a_1 a_0)$.

So, I decided to write out an example: $2^{16}$.

This is where is ran into some trouble. $65536$ is a five-digit number, but I can't write $10^5 \cdot 6$ as this would give me $600000$. Can someone please help me out and explain?

Answer 1

There must be a typo in your book: it should read "If $N$ is a $(k+1)$-digit number…". The rest is fine.

–Aretino

Answer 2

As Aretino comments, your book should say

If $N$ is a $(k+1)$-digit number with digits $a_k, a_{k-1}...a_2, a_1, a_0$, then
$$N = 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10^2 a_2 + 10 a_1 + a_0.$$

You can think of this as a way of "exploding" the number by expanding its expression, writing it as a sum of numbers each of which has only one nonzero digit; this makes more sense after seeing a few examples:

$$25= 20+5=10^1\cdot2+10^0\cdot 5 \\ 343 = 300+40+3=10^2\cdot 3 + 10^1\cdot 4+ 10^0\cdot 3 \\ 2048 = 2000+000+40+8 = 10^3\cdot 2 + 10^2\cdot 0 + 10^1\cdot 4+10^0\cdot 8 $$

Now, for your example, $$65535 = 60000+5000+500+30+5 = 10^4\cdot 6+ 10^3\cdot 5+ 10^2\cdot 5+ 10^1\cdot 3+10^0\cdot 5$$