query: The divisibility of the product of Pythagorean Theorem by 5
positive_passages.docid: 142133
positive_passages.text: Prove that if $a^2 + b^2 = c^2$, then 5 divides abc. a,b,c are positive integers.
negative_passages.docid: 42526
negative_passages.text: The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)
negative_passages.docid: 125210
negative_passages.text: proof: primitive pythagorean triple, a or b has to be divisible by 3
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
negative_passages.docid: 95141
negative_passages.text: $n^5-n$ is divisible by $10$?
negative_passages.docid: 10589
negative_passages.text: How to factorize $5$ in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 25910
negative_passages.text: Product of Quadratic Residues
negative_passages.docid: 150528
negative_passages.text: Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
negative_passages.docid: 116363
negative_passages.text: If $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$ , then prove that $P(x)$ is divisible by $x-1$
negative_passages.docid: 3611
negative_passages.text: How to prove $n^5 - n$ is divisible by $30$ without reduction
negative_passages.docid: 31555
negative_passages.text: Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
negative_passages.docid: 85959
negative_passages.text: How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3$?
negative_passages.docid: 130615
negative_passages.text: divisibility of binomial coefficients by primes
negative_passages.docid: 151520
negative_passages.text: About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
negative_passages.docid: 13478
negative_passages.text: Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
negative_passages.docid: 82071
negative_passages.text: divisor of a product of integers is a product of divisors
negative_passages.docid: 152231
negative_passages.text: Computing the product of p/(p - 2) over the odd primes