Do powers of 256 all end by 6 and if so, how to prove it?

$$(10x+6)(10y+6)=100xy+60 (x+y)+3\color {red}6. $$


In $\mathbb{Z}_{10}$ (the integers moduo $10$) we have that $6^2 = 6$, an idempotent, so all its positive powers are $6$ too. And $256 \equiv 6 \pmod{10}$.


Let $P(n)$ be the statement that $6^n$ ends on $6$.

Then evidently $P(1)$ is true.

If $P(n)$ is true then it is not difficult to prove that also $P(n+1)$ is true (try it yourself).

According to the principle of induction on natural numbers we are allowed to conclude that $P(n)$ is true for every positive integer.

Tags:

Exponentiation

Modular Arithmetic

Proof Writing

Related

The average radius of an ellipse Find positive $K$ such that $\int_0^\infty\left(\frac{1}{\sqrt{2x^2+1}}-\frac{K}{x+1}\right)dx$ converges What is the series expansion of $ \sqrt[x] x $? Evalulate $\int_0^\infty \int_0^\infty \int_0^\infty e^{-(xy+yz+zx)}\ dx\ dy\ dz$ If a collection of closed sets of arbitrary cardinality in a metric space has empty intersection, does some countable subcollection? Do we know the limit of $\sum\limits_{k=1}^{n}\frac{1}{(a i k+b)^2}$? Important Olympiad-inequalities Convergence of $a_{n+2}^3+a_{n+2}=a_{n+1}+a_n$ Is the normality of a subgroup dependent on which group is its parent? Will a path between $(x, y)$ and $(-x, -y)$ always intersect a 90 degree rotated copy? Why is Catalan's constant $G$ important? Compute Integral $\int _{0}^{\infty} \frac {(\ln x)^2}{\sqrt x (1-x)^2} \mathrm dx$

Recent Posts