query: If $a,b,c \in \mathbb{Z}$ and $a^2+b^2=c^2$ show that $abc \equiv 0 \pmod{60}$.
positive_passages.docid: 152167
positive_passages.text: Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.
negative_passages.docid: 95162
negative_passages.text: For a odd prime $p, \exists a,b$ such that $a^2 + ab + b^2 \equiv 0 \pmod{p} \iff \exists x,y $ such taht $x^2 + xy + y^2 = p$?
negative_passages.docid: 100426
negative_passages.text: Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
negative_passages.docid: 24820
negative_passages.text: Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
negative_passages.docid: 91113
negative_passages.text: $ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $
negative_passages.docid: 110256
negative_passages.text: Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
negative_passages.docid: 70619
negative_passages.text: Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.
negative_passages.docid: 49548
negative_passages.text: If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
negative_passages.docid: 18203
negative_passages.text: Diophantine equation $a^2+b^2=c^2+d^2$
negative_passages.docid: 36043
negative_passages.text: If $ab \equiv r \pmod{p}$, and $x^2 \equiv a \pmod{p}$ then $y^2 \equiv b \pmod{p}$ for which condition of $r$?
negative_passages.docid: 29640
negative_passages.text: If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
negative_passages.docid: 66161
negative_passages.text: How to solve the equation $n^2 \equiv 0 \pmod{584}$?
negative_passages.docid: 51153
negative_passages.text: Number of solutions to $x^2\equiv a\pmod m$
negative_passages.docid: 27544
negative_passages.text: With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
negative_passages.docid: 97750
negative_passages.text: Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
negative_passages.docid: 61987
negative_passages.text: $x^2+1=0$ in $\mathbb{Z}_7$
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.