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In fact, this rule for prime divisors besides 2 and 5 is really a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below). This is why the last divisibility condition in the tables above and below for any number relatively prime to 10 has the same kind of form (add or subtract some multiple of the ...
The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10. In mathematics, a divisor of an integer , also called a factor of , is an integer that may be multiplied by some integer to produce . [1] In this case, one also says that is a multiple of .
Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called fields leaves division by zero undefined , and situations where division by zero might ...
Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). A magma for which both a \ b and b / a exist and are unique for all a and all b (the Latin square property) is a quasigroup. In a quasigroup, division in this sense is always possible, even without an ...
The following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic ...
The least common multiple of a and b is equal to their product ab, i.e. lcm(a, b) = ab. [4] As a consequence of the third point, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a). [5] That is, we may "divide by b" when working modulo a.
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Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations.