Cubes and Cube Roots
Hardy – Ramanujan Numbers like 1729, 4104, 13832. They can be expressed as sum of two cubes in two different ways. e.g. 1729 = 1728 + 1 = 123 + 13 1729 = 1000 + 729 = 103 + 93 Cube numbers: The numbers obtained by multiplying another number three times with itself are called cube numbers or simply cubes. If a and b are two natural numbers such that a3 = b, then b is called the cube of a. If the units digit of a3 is b, then the cubes of all numbers ending with a will have their units digit as b. Properties of cube of natural numbers The cubes of all numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7 as the units digit. The cubes of all numbers that end in digits 1, 4, 5,6 and 9 are the numbers ending in the same digit. The sum of the cubes of first n naturals is equal to the square of their sum. Cubes of all even natural numbers are even. Cubes of all odd natural numbers are odd. Method to find the cube of a two digit number 1) Column method. The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on. •  If a given number is a perfect cube, then its prime factors will always occur in groups of three. Cube root A number a is the cube root of a number b if b = a3. e.g. ∛512 = 8 . Here ∛ is an cube root symbol. The cube root of a number can be found using the prime factorisation method or estimation method.

#### Summary

Hardy – Ramanujan Numbers like 1729, 4104, 13832. They can be expressed as sum of two cubes in two different ways. e.g. 1729 = 1728 + 1 = 123 + 13 1729 = 1000 + 729 = 103 + 93 Cube numbers: The numbers obtained by multiplying another number three times with itself are called cube numbers or simply cubes. If a and b are two natural numbers such that a3 = b, then b is called the cube of a. If the units digit of a3 is b, then the cubes of all numbers ending with a will have their units digit as b. Properties of cube of natural numbers The cubes of all numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7 as the units digit. The cubes of all numbers that end in digits 1, 4, 5,6 and 9 are the numbers ending in the same digit. The sum of the cubes of first n naturals is equal to the square of their sum. Cubes of all even natural numbers are even. Cubes of all odd natural numbers are odd. Method to find the cube of a two digit number 1) Column method. The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on. •  If a given number is a perfect cube, then its prime factors will always occur in groups of three. Cube root A number a is the cube root of a number b if b = a3. e.g. ∛512 = 8 . Here ∛ is an cube root symbol. The cube root of a number can be found using the prime factorisation method or estimation method.