query: Show that $19\mid 12^{2013}+7^{2013}$
positive_passages.docid: 153892
positive_passages.text: Number Theory divisibilty
negative_passages.docid: 147140
negative_passages.text: Prove the divisibility test by $7,11,13$ for numbers more than six digits
negative_passages.docid: 144447
negative_passages.text: Show that $504 \mid (n^9 − n^3 )$ for any integer $n$
negative_passages.docid: 6777
negative_passages.text: Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 81058
negative_passages.text: Evaluate $6^{433} \pmod {21}$ and a proving question
negative_passages.docid: 64592
negative_passages.text: The last two digits of $9^{9^9}$
negative_passages.docid: 92700
negative_passages.text: How can I solve $x^2 \equiv 19 \pmod {59}$.
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$
negative_passages.docid: 117969
negative_passages.text: Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$.
negative_passages.docid: 145089
negative_passages.text: Inequality: $7a+5b+12ab\le9$
negative_passages.docid: 127293
negative_passages.text: Prove that $(a-b) \mid (a^n-b^n)$
negative_passages.docid: 17609
negative_passages.text: Finding the remainder of $11^{2013}$ divided by $61$
negative_passages.docid: 68960
negative_passages.text: How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
negative_passages.docid: 126888
negative_passages.text: How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
negative_passages.docid: 59135
negative_passages.text: What is the proof of divisibility by $13$?
negative_passages.docid: 69631
negative_passages.text: Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$
negative_passages.docid: 14139
negative_passages.text: $(12)$ and $(123\dots n)$ are generators of $S_n$