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The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd. All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form a 2 for some a ).
That 641 is a factor of F 5 can be deduced from the equalities 641 = 2 7 × 5 + 1 and 641 = 2 4 + 5 4. It follows from the first equality that 2 7 × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2 28 × 5 4 ≡ 1 (mod 641).
= 2.731 177 3 × 10 −7 m 3 /s cubic inch per second in 3 /s ≡ 1 in 3 /s = 1.638 7064 × 10 −5 m 3 /s: cubic metre per second (SI unit) m 3 /s ≡ 1 m 3 /s = 1 m 3 /s gallon (US fluid) per day GPD [citation needed] ≡ 1 gal/d = 4.381 263 63 8 × 10 −8 m 3 /s gallon (US fluid) per hour GPH [citation needed] ≡ 1 gal/h = 1.051 503 27 3 ...
For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2 3 × 3 2 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic. [5] [6] [7] [8]
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
The elements 2 and 1 + √ −3 are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √ −3, but they are not associated, so there is no greatest common divisor of a and b.
Also, the product of neighboring digits 3 × 4 is 12, like 4 × 3, while the sum of its prime factors 7 + 7 + 7 is 21. 307 has a prime index of 63, or thrice 21: 3 × 3 × 7, equivalently 3 × 7 × 3 and 7 × 3 × 3, are all permutations of the prime factorization of 21.
483: 48 − (3 × 9) = 21 = 7 × 3. Adding 3 times the first digit to the next and then writing the rest gives a multiple of 7. (This works because 10a + b − 7a = 3a + b; the last number has the same remainder as 10a + b.) 483: 4 × 3 + 8 = 20, 203: 2 × 3 + 0 = 6, 63: 6 × 3 + 3 = 21. Adding the last two digits to twice the rest gives a ...