# Divisibility 2 – The Block Split Method

The Block Split Method

To derive the test of divisibility for n (a natural number), say we have this relation –

$10^r \equiv\ k (mod n)$

Let us take a natural number $\underbrace{a_la_{l-1}\hdots a_{l-r}}_x\underbrace{a_{l-r-1}\hdots a_2a_1}_y$. When will $n | a_la_{l-1}\hdots a_2a_1$? –

Interpret the number taken this way –

$latex a_{l}a_{l-1}\hdots a_2a_1= x\cdot 10^r +y$

If,

$n|a_la_{l-1}\hdots a_2a_1$

$\implies n|10^rx+y$

$\implies kx+y \equiv\ 0 (mod n)$

So we are done if the last congruence relation alone is checked.

Let us try applying this for certain simple cases –

Check for divisibility for 1001

Note that $10^3 \equiv\ -1 (mod 1001)$

Let the number be $\underbrace{XYZ}_p\underbrace{ABC}_q$

Interpret the number as –

$XYZABC = 10^3\cdot p + q$

$latex 1001p + q – p$

Thus, essentially we can check the divisibility of the number XYZABC by checking the same for – $ABC - XYZ$.