query: Why is any number N of base B, whose digits sum to a multiple of a prime factor P of B-1, also divisible by P?
positive_passages.docid: 78946
positive_passages.text: Why does casting out $9$ and $11$ work to compute remainders?
negative_passages.docid: 11470
negative_passages.text: Divisibility of integers
negative_passages.docid: 75516
negative_passages.text: Relatively Prime Numbers and Number of Divisors of n
negative_passages.docid: 17194
negative_passages.text: $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
negative_passages.docid: 24053
negative_passages.text: How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
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: 130842
negative_passages.text: Prime factors + number of Divisors
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: 138645
negative_passages.text: Prove Divisibility In Fibonacci Sequence Over A Prime Number
negative_passages.docid: 116970
negative_passages.text: On the number divisors of a number
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: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 32618
negative_passages.text: Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
negative_passages.docid: 153892
negative_passages.text: Number Theory divisibilty
negative_passages.docid: 143620
negative_passages.text: How many $n$-digit decimal sequences (using the digits $0 = 9$) are there in which the digits $1$, $2$ and $3$ all appear?
negative_passages.docid: 49687
negative_passages.text: Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9.
negative_passages.docid: 41563
negative_passages.text: Prove that if the sum of digits of a number is divisible by 3, so is the number itself.