Basic Modular Exponentiation question

Question

I just know this rule :ab mod n = [(a mod n)(b mod n)] mod n. How can it be proved that the following rule is true ?

x^7 mod n = {(x^4 mod n)(x^2 mod n)(x^1 mod n)} mod n

Answer 1

The basic idea of congruences is to say that one number is equivalent to another by a relation. Numbers that are equivalent are said to belong to the same equivalence class.

In the case of mod, the equivalence relation is "leave the same remainder when divided by n". So if we consider for example 22 and 48 mod 13, they both leave remainder 9. $22=13+9$, and $48=3.13+9$, and so 22 and 48 belong to the equivalence class of mod13. We write $22\equiv48 \mod 13$. We usually simplify the right hand number and write $22\equiv9\mod13$.

This allows us to express general numbers in terms of the moduli (13 from the example just given), and a remainder.

In your example, the moduli is $n$, and we can write $a$ and $b$ as:

• $a=jn+x$
• $b=kn+y$

$a\equiv x \mod n$, and $b\equiv y \mod n $.

$ab=(jn+x)(kn+y)=jkn^2+n(jy+kx)+xy = n(jkn+jy+kx)+xy$. The remainder when divided by n is therefore $xy$, and we write $ab\equiv xy \mod n$, and as this the same as $(a \mod n)(b \mod n) \mod n $, we have proved your result.

To prove the $x^7$ result, use $x^7=x^4x^3$ and apply the theorem, and then reapply the theorem to $x^3=x^2x$.