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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. If any number ends in a two digit number that ...
Cuisenaire rods: 5 (yellow) cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) can be evenly divided in 2 by 3 (lime green). In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. [1]
A sanity test can refer to various orders of magnitude and other simple rule-of-thumb devices applied to cross-check mathematical calculations. For example: If one were to attempt to square 738 and calculated 54,464, a quick sanity check could show that this result cannot be true. Consider that 700 < 738, yet 700 2 = 7 2 × 100 2 = 490,000 ...
Two properties of 1001 are the basis of a divisibility test for 7, 11 and 13. The method is along the same lines as the divisibility rule for 11 using the property 10 ≡ -1 (mod 11). The two properties of 1001 are 1001 = 7 × 11 × 13 in prime factors 10 3 ≡ -1 (mod 1001) The method simultaneously tests for divisibility by any of the factors ...
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if > > are coprime integers, then for any integer , there is a prime number p (called a primitive prime divisor) that divides and does not divide for any positive integer <, with the following exceptions:
For example, in the case of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 × 15 = 60, but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. [note 2] It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings.
For example, if a = 2 and p = 7, then 2 7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] ().
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.