840 (number)

Template:Infobox number 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 $(\mathbb{Z}/840\mathbb{Z}{)}^{\times }\cong C_2\times C_2\times C_2\times C_2\times C_{12}$). Equivalently, it is the least modulus with 2⁵ square roots of 1.

References

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