840 (eight hundred
ndforty) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following 839 and preceding 841.
Mathematical properties
*It is an
even number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers.
The ...
.
*It is a
practical number.
*It is a
congruent number.
*It is the 15th
highly composite number
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(' ...
, with 32 divisors: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840. Since the sum of its divisors (excluding the number itself) 2040 > 840
*It is an
abundant number
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...
and also a
superabundant number.
*It is an
idoneal number.
*It is the
least common multiple
In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...
of the numbers from 1 to 8.
*It is the smallest number divisible by every natural number from 1 to 10, except 9.
*It is the number under 1000 with the most divisors, at 32.
*It is the largest number ''k'' such that all coprime
quadratic residues
In number theory, an integer ''q'' is a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; that is, if there exists an integer ''x'' such that
:x^2\equiv q \pmod.
Otherwise, ''q'' is a quadratic nonresidue mod ...
modulo ''k'' are squares. In this case, they are 1, 121, 169, 289, 361 and 529.
*It is an
evil number.
*It is a
palindrome number and a
repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".
Ex ...
number repeated in the positional numbering system in base 29 (SS) and in that in base 34 (OO).
*It is the sum of a twin prime (419 + 421).
*It is the triple-digit number with the most divisors at 32.
References
Integers
{{Num-stub