Overview
784 is a positive integer that sits between 783 and 785. It is a composite, even number and a perfect square: 784 = 28^2. Because 28 itself is a well-known perfect number (the second perfect number), 784 is the square of a perfect number, a property that is simple to state though relatively uncommon in casual number lists.
Factorization and divisors
Its prime factorization is 784 = 2^4 * 7^2. From this factorization one can derive its complete set of positive divisors. The divisor function gives it 15 positive divisors in all; these are:
- 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
Typical factor pairs include (1,784), (2,392), (4,196), (7,112), (8,98) and (14,56).
Arithmetic functions
Several standard multiplicative functions take simple values at 784. The divisor-count function τ(784) = 15. The sum of all divisors σ(784) = 1,767, so the sum of proper divisors is 1,767 − 784 = 983; because this exceeds 784, the number is classified as abundant. Euler's totient is φ(784) = 336.
Representations and bases
As a square, its integer square root is 28. In base representations: binary 1100010000, octal 1420, and hexadecimal 0x310. In Roman numerals 784 is written DCCLXXXIV. Viewed arithmetically, any square n^2 is the sum of the first n odd numbers, so 784 equals the sum of the first 28 odd integers.
Notable properties and remarks
The condition that each prime congruent to 3 mod 4 appears with an even exponent guarantees representation as a sum of two squares; here 7 ≡ 3 (mod 4) appears squared, so 784 can be expressed as a sum of two integer squares (trivially as 28^2 + 0^2, and also in nontrivial ways). Its structure 2^4*7^2 makes many arithmetic properties easy to compute by multiplicativity. As the square of a perfect number, 784 connects two familiar classes of integers (perfect numbers and squares) but has no exceptional status such as primality or being a perfect power beyond its square nature.