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6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7. 6174 can be written as the sum of the first three powers of 18: 18 3 + 18 2 + 18 1 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18. The sum of squares of the prime factors of 6174 is a square: 2 2 + 3 2 + 3 2 + 7 2 + 7 2 + 7 2 = 4 + 9 + 9 + 49 + 49 ...
Imaginary numbers: Numbers that equal the product of a real number and the imaginary unit , where =. The number 0 is both real and imaginary. The number 0 is both real and imaginary. Complex numbers ( C {\displaystyle \mathbb {C} } ): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8.
Consider the number 3.1400010000000000000000050000.... 3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2(4319 zeros)6... where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of π.
Another way of stating Carmichael's conjecture is that, if A(f) denotes the number of positive integers n for which φ(n) = f, then A(f) can never equal 1. Relatedly, Wacław Sierpiński conjectured that every positive integer other than 1 occurs as a value of A( f ), a conjecture that was proven in 1999 by Kevin Ford.
De Bruijn's theorem concerns the impossibility of packing certain cuboids into a larger cuboid. For instance, it is impossible, according to this theorem, to fill a 6 × 6 × 6 box with 1 × 2 × 4 cuboids. The proof uses a similar chessboard-coloring argument to the mutilated chessboard problem. [40]
For example, in duodecimal, 1 / 2 = 0.6, 1 / 3 = 0.4, 1 / 4 = 0.3 and 1 / 6 = 0.2 all terminate; 1 / 5 = 0. 2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1 / 7 = 0. 186A35 has period 6 in duodecimal, just as it does in decimal. If b is an integer base ...
The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero. For base b {\displaystyle b} and natural numbers k {\displaystyle k} and n > 9 {\displaystyle n>9} the numbers n {\displaystyle n} and n ⋅ b k {\displaystyle n\cdot b^{k}} have the same additive persistence.