the number $1729$ is known as Ramanujan's number. The great indian number theoriest found two different ways to write $1729$ as the sum of two cubes. The first way is $12^3+1^3=1729$ . what is the second way

View step-by-step help

Home

Math Problems

Algebra 1

Evaluate expressions using exponent rules

Full solution

Q. the number

1729

is known as Ramanujan's number. The great indian number theoriest found two different ways to write

1729

as the sum of two cubes. The first way is

12^3+1^3=1729

. what is the second way

Understand the problem: Understand the problem. $\newline$ We need to find another pair of cubes that add up to $1729$ , different from $12^3 + 1^3$ .
Use Ramanujan's property: Use the property of Ramanujan's number. $\newline$ Since $1729$ is known as a taxicab number, it can be expressed as the sum of two cubes in two distinct ways. We already have the first way: $12^3 + 1^3$ . We need to find the second way.
Find second pair of cubes: Find the second pair of cubes. $\newline$ We know that the cubes must be smaller than $12^3$ since $12^3$ is already one of the pairs. We can start by checking the cubes of numbers less than $12$ .
Check cubes of numbers: Check the cubes of numbers less than $12$ . We can start with $11$ and go down, checking if $(1729 - 11^3)$ is a perfect cube.
Calculate and subtract: Calculate $11^3$ and subtract it from $1729$ . $\newline$ $11^3 = 11 \times 11 \times 11 = 1331$ $\newline$ $1729 - 1331 = 398$
Check if perfect cube: Check if $398$ is a perfect cube. $\newline$ The cube root of $398$ is not an integer, so $11^3$ is not part of the second pair. We move on to the next integer.
Calculate and subtract: Calculate $10^3$ and subtract it from $1729$ . $\newline$ $10^3 = 10 \times 10 \times 10 = 1000$ $\newline$ $1729 - 1000 = 729$
Check if perfect cube: Check if $729$ is a perfect cube. $\newline$ The cube root of $729$ is $9$ , since $9^3 = 729$ .
Conclude second pair of cubes: Conclude the second way to express $1729$ as the sum of two cubes. $\newline$ The second way to express $1729$ as the sum of two cubes is $10^3 + 9^3$ .