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The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; [2][3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4.
The GCD of a and b is their greatest positive common divisor in the preorder relation of divisibility. This means that the common divisors of a and b are exactly the divisors of their GCD. This is commonly proved by using either Euclid's lemma, the fundamental theorem of arithmetic, or the Euclidean algorithm. This is the meaning of "greatest ...
Digital root. The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.
Euclid's lemma. In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: [note 1] Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019 ...
Number theory looks at things like how numbers divide evenly (divisibility), or how prime numbers are spread out. Combinatorics studies counting and arranging numbered objects, such as partitions and enumerations.
Goldbach's conjecture is used when studying computation complexity. [ 36 ] The connection is made through the Busy Beaver function, where BB (n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 27 state Turing machine that halts if and only if Goldbach's conjecture is false.
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