The book goes deep quite quickly, indeed.

At what chapter of with what kinds of calculations do you have problems?

I’ll try to make a post of it.

# De kunst van het hoofdrekenen (Art of Mental Calculation)

Thanks Kinma.

For example, in the chapter about **the Chinese Remainder** I have understand nothing. The only thing I get it’s Mr. Bouman use modular arithmetic, but even that I don’t understand

I agree. That part of Willems book is difficult to understand because he does not explain how he comes to the numbers he works with.

First, let’s show the kind of puzzle we are trying to solve:

(Source: Exploring Continued Fractions: From the Integers to Solar Eclipses

By Andrew J. Simoson)

Willems example is this. He uses different numbers though:

x = 2 (mod 3) = 3 (mod 4) = 5 (mod 7)

In plain English. I am looking for a number - or a range of numbers - when divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3 and also divided by 7 leaves a remainder of 5.

First he starts solving the first 2: x = 2 (mod 3) = 3 (mod 4)

He constructs 11, because 11 is both 2 (mod 3) and 3 (mod 4).

Then he constructs a way to find the answer.

The exact way he does this makes this post too difficult to read, so I’ll just copy a visual example of a solution to another problem from the Wikipedia page:

Sunzi’s used this example in his original work in the 4th century:

x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7)

with the solution x = 23 + 105k (where k is a natural number).

(source: https://en.wikipedia.org/wiki/Chinese_remainder_theorem)

The pirate example might seem far fetched.

An other example of the kinds of problems one can tackle with the theorem is the following.

Shameless copy:

Source: http://homepages.math.uic.edu/~leon/mcs425-s08/handouts/chinese_remainder.pdf

What is it that you don’t understand about modular arithmetic?

On this forum I use it a lot to show how I check my answers.

Here is a short primer that I wrote a long time ago: