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 a sparsely totient number It is a Harshad number in base 2 through base 10 It is a balanced number It is the 15th highly composite number, 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. It is an idoneal number. It is the least common multiple 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 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 number repeated in the positional numbering system in base 29 (SS) and 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. It is the smallest k such that the multiplicative group of integers modulo k is the direct product of 5 but no fewer cyclic groups. (Here ( Z / 840 Z ) × ≅ C 2 × C 2 × C 2 × C 2 × C 12 {\displaystyle (\mathbb {Z} /840\mathbb {Z} )^{\times }\cong C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}} ). Equivalently, it is the least modulus with 25 square roots of 1.
References