query: Looking for a theorem talking about the remainder when a number is divided by 9
positive_passages.docid: 3128
positive_passages.text: Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens)
negative_passages.docid: 28661
negative_passages.text: What will be the one's digit of the remainder in: $\left|5555^{2222} + 2222^{5555}\right|\div 7=?$
negative_passages.docid: 124295
negative_passages.text: Divison and remainder in Prolog
negative_passages.docid: 73020
negative_passages.text: Every integer is congruent to the sum of its digits mod 9
negative_passages.docid: 80502
negative_passages.text: $p(x)$ divided by $x-c$ has remainder $p(c)$? [Polynomial Remainder Theorem]
negative_passages.docid: 146203
negative_passages.text: if p is prime then 1920 divides $p^4-10p^2+9$
negative_passages.docid: 156022
negative_passages.text: How to perform an integer division, and separately get the remainder, in JavaScript?
negative_passages.docid: 52856
negative_passages.text: Taylor's Theorem with Peano's Form of Remainder
negative_passages.docid: 95033
negative_passages.text: Find the remainder of a number with a large exponent
negative_passages.docid: 72406
negative_passages.text: Last two digits of $2^{1000}$ via Chinese Remainder Theorem?
negative_passages.docid: 122878
negative_passages.text: How can I prove that $2/9<x^4+y^4<8$?
negative_passages.docid: 142151
negative_passages.text: Find the last two digits of $9^{{9}^{9}}$
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: 100336
negative_passages.text: Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
negative_passages.docid: 17243
negative_passages.text: Remainder when $2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by $1998$
negative_passages.docid: 99243
negative_passages.text: What is the remainder when $24^{1202}$ is divided by $1446$?
negative_passages.docid: 140141
negative_passages.text: Arithmetic of continued fractions, does it exist?