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The result must be divisible by 3. Using the example above: 16,499,205,854,376 has four of the digits 1, 4 and 7 and four of the digits 2, 5 and 8; Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3. Subtracting 2 times the last digit from the rest gives a multiple of 3. (Works because 21 is divisible by 3)
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and ...
if a number is divisible by 2, but not by 3, its square ends in 4, and its preceding digit must be 0, 1, 4, 5, 8, or 9; and; if a number is not divisible by 2, but by 3, its square ends in 9, and its preceding digit must be 0 or 6. Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for ...
Amicable numbers. Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s (a)= b and s (b)= a, where s (n)=σ (n)- n is equal to the sum of positive divisors of n except n itself (see also divisor function). The smallest pair of amicable ...
Engineering notation or engineering form (also technical notation) is a version of scientific notation in which the exponent of ten is always selected to be divisible by three to match the common metric prefixes, i.e. scientific notation that aligns with powers of a thousand, for example, 531×10 3 instead of 5.31×10 5 (but on calculator displays written without the ×10 to save space).
By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also -perfect numbers, or Granville numbers. A semiperfect number is a natural number that is equal to the sum of all or some of its ...
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.
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...