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The leading number is never 0

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Every positive integer n other than 1 is divisible by a prime.

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Numbers are congruent when they have the same remainder when divided by a number n.

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Every integer bigger than 1 can be factored into prime factors only 1 way

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All integers other than 1 can be divided by a prime.

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When solving a Diophantine equation, it is sometimes helpful to write the equation in the form of modular congruences.

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let a, b, c, and d be integers.

1. If a|band b|c, then a|c(transitivity of divisibility).

2. If a|b, then a|bc.

3. If a + b = c and if d divides two of a, b, and c, then d also divides the third.

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When translating letters to numbers in a mathematical operation such as addition...

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When translating letters to numbers in a mathematical operation such as addition...

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you start with the left most column and find the greatest possible integers for the letters that could equal the result

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m divides n if there is such a number q that n = mq is true.

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A diophantine equation is an equation in which all coefficients are integers and in which we are interested only in integer solutions. (A linear equation could be diophantine or not depending on whether it has integer solutions. If it does not, then it is not a diophantine equation.)

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You can use the prime factorization of a number to find all of its divisors.

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Places different number bases are used...

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Places different number bases are used...

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They use hybrid number bases in digital clocks and at baseball games, base twelve at stores to track merchandise, and even the English system of measurement uses different bases!

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numbers such as 8 that exceed the sum of their proper divisors: 8 > 1 + 2 + 4

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A composite number is a number other than 1 that is not a prime.

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numbers such as 6 that are equal to the sum of their proper divisors: 6 = 1 + 2 + 3

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our mathematical operations are essentially the same no matter the base

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Base ten

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A positive integer p that has exactly two divisors in N.

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You can find all primes between two numbers using this method.

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Cryptarithmetic problems where letters form words

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a problem where digits may be coded or partially missing and mathematical functions are needed to reconstruct the original calculation

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All numbers and most equations can be written in modular form. For example, 10 can be written as 2(modulo 8) because when 10 is divided by 8, the remainder is 2.

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There are primes, perfect numbers, and deficient numbers.