72 (number)

72 is a natural number that comes after 71 and before 73. It is an even number composed of two digits: 7 and 2. In mathematics, it is notable for having many divisors relative to numbers of similar size.

72 appears in many practical contexts: geometric progressions, financial calculations, and time measurement (72 hours = 3 days). It is also the atomic number of hafnium in the periodic table.

Mathematical properties

In prime factorization, 72 = 2³ × 3², which can also be written as 8 × 9. It has 12 divisors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The sum of its divisors less than 72 exceeds 72 itself, making 72 an abundant number (one whose proper divisors sum to more than the number itself).

The sum of the digits of 72 is 7 + 2 = 9, and 72 is divisible by 9. For this reason, 72 is a Harshad number (one divisible by the sum of its digits). Euler's totient function φ(72) is 24, meaning 24 numbers less than 72 share no common divisor with 72 other than 1.

Geometry

A circle divided into five equal parts creates a central angle of 72 degrees (360 ÷ 5 = 72). For this reason, 72° is associated with the regular pentagon and five-pointed star. The interior angle of a regular pentagon is 108°, which equals 180° − 72°.

Financial use: the Rule of 72

In finance and economics, the Rule of 72 is a simple method to estimate how many years it takes for money to double given a known annual rate of return. The number of years is approximated by dividing 72 by the percentage rate of return. For example, at 6% return, it would take roughly 12 years. This is an approximation, not an exact calculation.