Answer
Verified
423.3k+ views
Hint: Consider any three natural numbers and from this first find out the sum of the three consecutive natural numbers and then make use of the divisibility test and find out the answer.
Complete step-by-step answer:
Let us consider the three successive natural numbers to be a-1, a, a+1.
Now, we have to find out the cubes of these three consecutive numbers is divisible by what
So, the cubes of these three consecutive numbers would be \[{(a - 1)^3},{a^3},{(a + 1)^3}\]
Now, the sum of the cubes of these numbers would be
${(a - 1)^3} + {a^3} + {(a + 1)^3}$
Now , we know the formula which says ${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$
and ${\left( {a - b} \right)^3} = {a^3} + {b^3} - 3ab(a - b)$
So, making use of this formula , we can write
${(a - 1)^3} + {a^3} + {(a + 1)^3}$=${a^3} - 3{a^2} + 3a - 1 + {a^3} + {a^3} + 3{a^2} + 3a + 1$
= $3{a^3} + 6a$
=$3a({a^2} + 2)$
So, the sum of the cubes of three successive natural numbers
=$3a({a^2} + 2)$
In this equation either a or ${a^2} + 2$ has to be a multiple of 3
Now, from this we can clearly say that if a is multiple of 3 , then 3a is a multiple of 9
Else, if a is not a multiple of 3, then ${a^2} + 2$ is a multiple of 3
So, on combining these two results, we can say that $3a({a^2} + 2)$ is a multiple of 9 for all a$ \in $ N
So, from this we can say that the sum of cubes of 3 consecutive natural numbers is divisible by 9.
Note: The three consecutive natural numbers need not be a-1, a, a+1 only. We can take any other three consecutive natural numbers and solve it.
Complete step-by-step answer:
Let us consider the three successive natural numbers to be a-1, a, a+1.
Now, we have to find out the cubes of these three consecutive numbers is divisible by what
So, the cubes of these three consecutive numbers would be \[{(a - 1)^3},{a^3},{(a + 1)^3}\]
Now, the sum of the cubes of these numbers would be
${(a - 1)^3} + {a^3} + {(a + 1)^3}$
Now , we know the formula which says ${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$
and ${\left( {a - b} \right)^3} = {a^3} + {b^3} - 3ab(a - b)$
So, making use of this formula , we can write
${(a - 1)^3} + {a^3} + {(a + 1)^3}$=${a^3} - 3{a^2} + 3a - 1 + {a^3} + {a^3} + 3{a^2} + 3a + 1$
= $3{a^3} + 6a$
=$3a({a^2} + 2)$
So, the sum of the cubes of three successive natural numbers
=$3a({a^2} + 2)$
In this equation either a or ${a^2} + 2$ has to be a multiple of 3
Now, from this we can clearly say that if a is multiple of 3 , then 3a is a multiple of 9
Else, if a is not a multiple of 3, then ${a^2} + 2$ is a multiple of 3
So, on combining these two results, we can say that $3a({a^2} + 2)$ is a multiple of 9 for all a$ \in $ N
So, from this we can say that the sum of cubes of 3 consecutive natural numbers is divisible by 9.
Note: The three consecutive natural numbers need not be a-1, a, a+1 only. We can take any other three consecutive natural numbers and solve it.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE