# Tag Info

Accepted

### Are $14$ and $21$ the only "interesting" numbers?

Note that exactly one of $n$ and $n+1$ is even. It follows that for $n$ to be interesting, either $n=3p$ and $n+1=2N(p)$ or $n=2p$ and $n+1=3P(p)$, where $P(p)$ and $N(p)$ are the previous and next ...
Accepted

### Prove that the sum of pythagorean triples is always even

Note that $x^2\equiv x\pmod 2$ and thus $a^2+b^2=c^2$ implies $$a+b+c\equiv a^2+b^2+c^2\equiv 2c^2\equiv 0\pmod 2$$
Accepted

### What is the smallest positive multiple of 450 whose digits are all zeroes and ones?

Can we agree that it must be an even multiple of 450? Otherwise the last two digits will be 50. What is the smallest positive multiple of 900 such that all the digits are 0s or 1s? A rule of ...
Accepted

### How to prove that all odd powers of two add one are multiples of three

Since $2 \equiv -1 \pmod{3}$, therefore $2^{k} \equiv (-1)^k \pmod{3}$. When $k$ is odd this becomes $2^k \equiv -1 \pmod{3}$. Thus $2^k+1 \equiv 0 \pmod{3}$.