"This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8). Also of note is the modulus 11: since formula_35 and thus formula_36 taking the ""alternating"" sum of digits yields the value modulo 11. The digital root of a number is zero if and only if the number is itself zero. The digital root of a"
"the cube root of numbers having many digits in the ""Aryabhatiya"" (section 2.5). Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"""
"however, easy to show that most numbers are not perfect cubes because ""all"" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: The equation has"
"Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"" are The cube root operation is not distributive with addition or subtraction. In"
"is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero: There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above. Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit. The second step"
"cube root of 29791 is 31. Another example: This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11. These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work. To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. One must know: From"
"root is coprime with 10 are somewhat easier since the digits are scrambled in consistent ways, as we shall see in the next section. An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42 = 74088. Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10: Observe that there"
"equivalent because of the lack of fixed centers. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. This gives a total number of permutations of The entire number is 35 173 780 923 109 452 777 509 592 367 006 557 398 539 936 328 978 098 352 427 605 879 843 998 663 990 903 628 634 874 024 098 344 287 402 504 043 608 416 113 016 679 717 941 937 308 041 012 307 368 528 117"
"three if and only if the sum of its cubed digits are a multiple of three. By similar reasoning, all happy numbers when summing up odd powers (e.g. cubes, fifth powers, seventh powers, etc.) of their digits must have a remainder of 1 when dividing by 3. All numbers that are congruent to 2 (mod 3) end in either 371 or 407. The only positive whole numbers that are the sum of the cubes of their digits are 1, 153, 370, 371 and 407 . For higher powers, the density of happy numbers declines. Taking the sum of the fourth"
"888 (number) 888 (eight hundred [and] eighty-eight) is the natural number following 887 and preceding 889. 888 is a repdigit (a number all of whose digits are equal), and a strobogrammatic number (one that reads the same upside-down on a seven-segment calculator display). 888 = 700227072 is the smallest cube in which each digit occurs exactly three times, and the only cube in which three distinct digits each occur three times. 888 the smallest multiple of 24 whose digit sum is 24, and as well as being divisible by its digit sum it is divisible by all of its digits."
"by a factor of 24. This is because the 24 possible positions and orientations of the first corner are equivalent because of the lack of fixed centers. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. This gives a total number of permutations of The entire number is 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000"
"some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the ""principal cube root"", denoted with the radical sign . The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8 are 8, −4 + 4""i"", and −4 − 4""i"". The cube roots"
"this question in ""Scientific American"", Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the square root of the smaller of two real roots of the polynomial formula_7, which works out to approximately 1.007435. For formula_8, the optimal side length of the largest square in an formula_5-dimensional hypercube is either formula_10 or formula_11, depending on whether formula_5 is even or odd respectively. Prince Rupert's cube In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube,"
"9,223,372,036,854,775,807 The number 9,223,372,036,854,775,807 is the integer equal to 2 − 1. Its prime factorization is , which is equal to The number 9,223,372,036,854,775,807, equivalent to the hexadecimal value , is the maximum value for a 64-bit signed integer in computing. It is therefore the maximum value for a variable declared as a long integer (, , or ) in many programming languages running on modern computers. The presence of the value may reflect an integer overflow, or error. This value is also the largest positive signed address offset for 64-bit CPUs utilizing sign-extended memory addressing (such as the x86-64"
"real number is a positive real number, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5""i"" and −5""i"", where ""i"" represents a number whose square is . A cube root of a number ""x"" is a number ""r"" whose cube is ""x"": Every real number ""x"" has exactly one real cube root, written formula_4. For example, Every real number has two additional complex cube roots. Expressing the degree of an ""n""th root in its exponent form, as in formula_37, makes"
"in 64 bits depends on the integer representation used. With the two most common representations, the range is 0 through 18,446,744,073,709,551,615 (2 − 1) for representation as an (unsigned) binary number, and −9,223,372,036,854,775,808 (−2) through 9,223,372,036,854,775,807 (2 − 1) for representation as two's complement. Hence, a processor with 64-bit memory addresses can directly access 2 bytes (=16 exabytes) of byte-addressable memory. With no further qualification, a ""64-bit computer architecture"" generally has integer and addressing processor registers that are 64 bits wide, allowing direct support for 64-bit data types and addresses. However, a CPU might have external data buses or address"
"49 (number) 49 (forty-nine) is the natural number following 48 and preceding 50. Forty-nine is the square of seven The sum of the digits of the square of 49 (2401) is the square root of 49. 49 is the first square where the digits are squares. In this case 4 and 9 are squares. It appears in the Padovan sequence, preceded by the terms 21, 28, 37 (it is the sum of the first two of these). Along with the number that immediately derives from it, 77, the only number under 3 digits not having its home prime known (as"
"Base64 Base64 is a group of similar binary-to-text encoding schemes that represent binary data in an ASCII string format by translating it into a radix-64 representation. The term ""Base64"" originates from a specific MIME content transfer encoding. Each Base64 digit represents exactly 6 bits of data. Three 8-bit bytes (i.e., a total of 24 bits) can therefore be represented by four 6-bit Base64 digits. The particular set of 64 characters chosen to represent the 64 place-values for the base varies between implementations. The general strategy is to choose 64 characters that are common to most encodings and that are also"
"row 1. Subtracting 96431 from 163640, we're left with 67209. Appending a zero, we have 672090 to consider for the next cycle (with the partial result 485.1) The second digit after the decimal point is 6, as the biggest partial product less than 672090 is 578586 from row 6. The partial result is now 485.16, and so on. Extracting the square root uses an additional bone which looks a bit different from the others as it has three columns on it. The first column has the first nine squares 1, 4, 9, ... 64, 81, the second column has the"
"of a number ""x"" are the numbers ""y"" which satisfy the equation For any real number ""x"", there is ""one"" real number ""y"" such that ""y"" = ""x"". The cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number."
"Snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. Topologically, under its highest symmetry, [3,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with T symmetry), 24 regular tetrahedra, and 96 triangular pyramids. It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He"
"(43 + 47 + 53 + 59 + 61), balanced prime, Chen prime, Eisenstein prime with no imaginary part, strictly non-palindromic number, Bernoulli irregular prime, Euler irregular prime, Gaussian prime, full reptend prime, Solinas prime, Ramanujan prime. 264 = 2·3·11, Harshad number. If you take the sum of all 2-digit numbers you can make from 264, you get 264: 24 + 42 + 26 + 62 + 46 + 64 = 264. 132 and 396 share this property. 264 equals the sum of the squares of the digits of its own square in base 15. This property is shared with"
"142,857 142857, the six repeating digits of , 0., is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of , , , , or respectively. 142,857 is a Kaprekar number and a Harshad number (in base 10). If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating"
"Once again, find the row with the largest value less than the current partial remainder 5453. This time, it is the third row with 4089. The next digit of the square root is 3. Repeat the same steps as before and subtract 4089 from the current remainder 5453 to get 1364 as the next remainder. When you rearrange the board, notice that the second column of the square root bone is 6, a single digit. So just append 6 to the current number on the board 136 to set 1366 on the board. Repeat these operations once more. Now the"
"65 (number) 65 (sixty-five) is the natural number following 64 and preceding 66. Sixty-five is the 23rd semiprime and the 3rd of the form (5.q). It is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0. This number is the magic constant of 5 by 5 normal magic square: formula_1 This number is also the magic constant of n-Queens Problem for n = 5. 65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two ways, 65 = 8 + 1 = 7 + 4."
"pair of equidistant digits plus the square of the center digit (of the digits to the right of the colon). In a square root calculation the quotient digit set increases incrementally for each step. Consider the perfect square 2809 = 53. Use the duplex method to find the square root of 2,809. This method is applicable for finding the square root of formula_155 and converges best for formula_156. This, however, is no real limitation for a computer based calculation, as in base 2 floating point and fixed point representations, it is trivial to multiply formula_157 by an integer power of"
"off with increasing N at sufficient speed to have people believe that ; the largest number now known not to be a sum of four cubes is 7373170279850, and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, 1290740 and 7373170279850, respectively.) 13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that"
"was mapped into memory, and only of RAM (plus 4 KB in between the ROMs) were available at startup. Most ""breadbox"" Commodore 64s used 4164 DRAM, with eight chips to total up 64K of system RAM. Late breadbox models and all C64Cs used 41464 DRAM (64K×4) chips which stored 32 KB per chip, so only two were required. Since 4164 DRAMs are 64K×1, eight chips are needed to make an entire byte, and the computer will not function without all of them present. Thus, the first chip contains Bit 0 for the entire memory space, the second chip contains Bit"
"64-bit computing In computer architecture, 64-bit computing is the use of processors that have datapath widths, integer size, and memory address widths of 64 bits (eight octets). Also, 64-bit computer architectures for central processing units (CPUs) and arithmetic logic units (ALUs) are those that are based on processor registers, address buses, or data buses of that size. From the software perspective, 64-bit computing means the use of code with 64-bit virtual memory addresses. However, not all 64-bit instruction sets support full 64-bit virtual memory addresses; x86-64 and ARMv8, for example, support only 48 bits of virtual address, with the remaining"
"24! × 12! × 2 possibilities for the inner and outer edges together. This gives a total number of permutations of The full number is precisely 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 possible permutations (about 283 duodecillion on the long scale or 283 tresvigintillion on the short scale). Some variations of the Professor's Cube have one of the center pieces marked with a logo, which can be put into four different orientations. This increases the number of permutations by a factor"
"Square root In mathematics, a square root of a number ""a"" is a number ""y"" such that ; in other words, a number ""y"" whose ""square"" (the result of multiplying the number by itself, or ) is ""a"". For example, 4 and −4 are square roots of 16 because . Every nonnegative real number ""a"" has a unique nonnegative square root, called the ""principal square root"", which is denoted by , where √ is called the ""radical sign"" or ""radix"". For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is"
"24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C, icositetrachoron, octaplex (short for ""octahedral complex""), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells. The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a"
"property that when represented as a sum of two prime numbers (per Goldbach's conjecture) both of the primes must be greater than or equal to 23. There are exactly 220 different ways of partitioning 64 = 8 into a sum of square numbers. It is a tetrahedral number, the sum of the first ten triangular numbers, and a dodecahedral number. If all of the diagonals of a regular decagon are drawn, the resulting figure will have exactly 220 regions. It is the sum of the sums of the divisors of the first 16 positive integers. The number 220 can also"