query: Every positive integer can be expressed uniquely as $N = a_0 + a_1 2 + a_n 2^n $ and all $ a_i $ belongs to $\{0, 1\}$
positive_passages.docid: 16235
positive_passages.text: Prove by induction that every positive integer is represented by some binary number?
negative_passages.docid: 125707
negative_passages.text: A simple proof (hopefully) for $2^x \neq 0$
negative_passages.docid: 33786
negative_passages.text: Proof that a certain fraction is always an integer
negative_passages.docid: 77200
negative_passages.text: Prove:that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014
negative_passages.docid: 23544
negative_passages.text: Complete induction proof that every $n > 1$ can be written as a product of primes
negative_passages.docid: 123195
negative_passages.text: Prove by induction that a^n+a^-n is an integer.
negative_passages.docid: 59459
negative_passages.text: Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality
negative_passages.docid: 59535
negative_passages.text: Prove that there exists such a set of $n$ positive integers
negative_passages.docid: 35874
negative_passages.text: Show that when $BA = I$, the solution of $Ax=b$ is unique
negative_passages.docid: 94855
negative_passages.text: Prove if a (mod n) has a multiplicative inverse, then it's unique
negative_passages.docid: 33810
negative_passages.text: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
negative_passages.docid: 67208
negative_passages.text: Can we say that there exist an integer $n$ such $A+nB$ invertible?
negative_passages.docid: 32463
negative_passages.text: $ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
negative_passages.docid: 129163
negative_passages.text: Showing that every positive integer can be represented in this form
negative_passages.docid: 105985
negative_passages.text: Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers
negative_passages.docid: 18108
negative_passages.text: Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
negative_passages.docid: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$