84 (number)

Overview

Eighty-four is a natural number that follows eighty-three and precedes eighty-five. It is an even, composite integer often encountered in elementary arithmetic, combinatorics and computer representations.

Factors and divisors

The prime factorization of 84 is 2^2 × 3 × 7. From that factorization one can list all positive divisors explicitly. These divisors are:

• 1
• 2
• 3
• 4
• 6
• 7
• 12
• 14
• 21
• 28
• 42
• 84

There are twelve positive divisors in total. The sum of the proper divisors (all divisors except the number itself) is greater than 84, so 84 is classified as an abundant number.

Representations and combinatorial role

In common positional bases 84 appears as 1010100 in binary and as 54 in hexadecimal. In the ASCII character set the decimal value 84 corresponds to the uppercase letter T. In combinatorics 84 is a familiar binomial coefficient: it equals C(9,3), the number of ways to choose three elements from nine.

Uses and notable facts

Numbers like 84 arise in counting problems, factorization exercises and algorithmic contexts where divisibility by small primes matters. Its factor structure (2^2·3·7) makes it useful in examples illustrating least common multiples, divisor functions and modular arithmetic. Because it has twelve divisors and is abundant, it often appears in elementary number-theory tables and classroom examples.

Summary

The integer 84 is a composite even number with a compact prime factorization and a moderate number of divisors. It serves as a clear example in teaching divisibility, representations (binary and hexadecimal) and simple combinatorial identities. For more references and numerical tables see related entries linked here: neighboring numbers, successor and other number-theory topics.