Proving Divisibility: Modular Arithmetic and the Pattern of 16^43 - 10^26 Mod 21

[The Subject]

Hi I'm reading a text about modular arithmetic,

Prove that 16^43 - 10^26 actually is divisible by 21.
They separate it by showing it is divisible by 7 and 3

they showed $$16 \equiv 2 \textrm{ mod 7} \\ <br /> 16^2 \equiv 2^2 \equiv 4 \textrm{ mod 7} \\<br /> 16 \equiv 2^3 \equiv 1 \textrm{ mod 7} \\$$
So there is a pattern of length 3.

They later made 43 = 3 * 14 +1 . so,
$$16^{43} \equiv 16^1 \equiv 2 \textrm{ mod 7} \\$$

whats the reasoning with 43 = 3 * 14 + 1 ?

 

[ehild]

Hi I'm reading a text about modular arithmetic,

Prove that 16^43 - 10^26 actually is divisible by 21.
They separate it by showing it is divisible by 7 and 3

they showed $$16 \equiv 2 \textrm{ mod 7} \\<br /> 16^2 \equiv 2^2 \equiv 4 \textrm{ mod 7} \\<br /> 16 ^3\equiv 2^3 \equiv 1 \textrm{ mod 7} \\$$
So there is a pattern of length 3.

They later made 43 = 3 * 14 +1 . so,
$$16^{43} \equiv 16^1 \equiv 2 \textrm{ mod 7} \\$$

whats the reasoning with 43 = 3 * 14 + 1 ?

$16^{43}=16^{42}\cdot 16 = (16^3)^{14}\cdot16$. You know that 16^3= 1 mod 7, 16=2 mod 7.