single-digit remains, which is called the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive integers...
6 KB (965 words) - 03:45, 22 January 2023
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing...
13 KB (2,523 words) - 07:08, 8 March 2024
Persistence of a number (redirect from Multiplicative Persistence)
is the smallest number of multiplicative persistence 3. In base 10, there is thought to be no number with a multiplicative persistence greater than 11;...
5 KB (685 words) - 10:15, 31 October 2024
Triangular number (redirect from Triangular root)
by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every...
25 KB (3,594 words) - 10:54, 2 June 2025
prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo pn (that is, the group of units of...
4 KB (601 words) - 21:33, 5 December 2024
decimal digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots...
6 KB (875 words) - 06:50, 10 February 2025
from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings...
117 KB (14,179 words) - 21:25, 8 June 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
5 KB (730 words) - 19:47, 12 December 2024
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
8 KB (658 words) - 12:08, 2 June 2025
Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine...
5 KB (1,028 words) - 05:35, 18 May 2025
Natural number (section Multiplication)
the real numbers add all infinite decimals. Complex numbers add the square root of −1. This chain of extensions canonically embeds the natural numbers in...
53 KB (5,889 words) - 09:45, 17 June 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
11 KB (1,215 words) - 05:31, 1 May 2025
terms in this formula are multiplicative but not completely multiplicative. Therefore, the harmonic mean H(n) is also multiplicative. This means that, for...
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invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1...
105 KB (13,693 words) - 08:02, 16 June 2025
includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However...
87 KB (13,080 words) - 23:42, 12 June 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
6 KB (851 words) - 22:15, 14 June 2025
congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide ( 2 p + 1 ) − 1 2 = p {\textstyle...
71 KB (6,408 words) - 19:11, 6 June 2025
perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's cube can...
24 KB (3,032 words) - 16:41, 16 May 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
4 KB (581 words) - 19:51, 23 April 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
11 KB (2,049 words) - 16:16, 13 May 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
7 KB (899 words) - 16:12, 16 June 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
8 KB (787 words) - 10:47, 24 December 2024
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
5 KB (447 words) - 19:37, 23 May 2025
→ 0 c ( x ) = 1 {\displaystyle C_{0}=\lim _{x\to 0}c(x)=1} . The square root term can be expanded as a power series using the binomial series 1 − 1 −...
40 KB (6,013 words) - 02:24, 6 June 2025
starting point 2k, and the period is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicative group of integers modulo n).[citation...
44 KB (4,372 words) - 22:09, 12 June 2025
Narcissistic number (redirect from Pluperfect Digital Invariant)
In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus...
10 KB (1,082 words) - 22:07, 2 February 2025
one as follows: Inserting zeroes into N will not change the sequence of digital sums (just as 21, 201 and 2001 are all 10-harshad numbers). If we insert...
18 KB (2,776 words) - 11:20, 10 April 2025
until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2...
38 KB (5,172 words) - 17:00, 3 June 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
7 KB (817 words) - 16:44, 24 March 2025
Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related Meertens Other Dudeney Factorion...
6 KB (846 words) - 20:22, 12 December 2024