840 (number) explained

Number:	840
Divisor:	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

840 (eight hundred [and] forty) is the natural number following 839 and preceding 841.

Mathematical properties

• It is an even number.
• It is a practical number.
• It is a congruent number.
• It is the 15th highly composite number,^[1] 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 and also a superabundant number.^[2]
• It is an idoneal number.^[3]
• It is the least common multiple of the numbers from 1 to 8.^[4]
• It is the smallest number divisible by every natural number from 1 to 10, except 9.
• It is the largest number k such that all coprime quadratic residues modulo k are squares. In this case, they are 1, 121, 169, 289, 361 and 529.^[5]
• It is an evil number.
• It is a palindrome number and a repdigit 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).

Notes and References

1. Highly composite numbers, definition (1): where d(n), the number of divisors of n (A000005), increases to a record.
2. Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m
3. Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers).
4. Least common multiple (or LCM) of for n >= 1.
5. Numbers k such that all coprime quadratic residues modulo k are squares..