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If ‘a’, ‘b’, ‘c’ and ‘d’ are consecutive natural numbers and \[{a^3} = {b^3} + {c^3} + {d^3}\], what is the least value of ‘a’?

Last updated date: 20th Jun 2024
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Hint: In the given question, we have been given that there are four consecutive natural numbers. And that, the sum of cubes of three natural numbers is equal to the cube of the fourth natural number. We have to find the least possible value of that fourth natural number. For solving this, we will substitute the value of all of the given numbers in the form of one natural number. Then, we are going to put in the possible values and once we reach a point where our condition is met, we stop and that is going to give us our answer.

Complete step-by-step answer:
For the given system to work, \[{a^3} = {b^3} + {c^3} + {d^3}\] is possible for consecutive natural numbers only if \[a > b > c > d\].
So, let us substitute the values of \[b,c,d\] in form of \[a\], and we get:
\[b = a - 1\]
\[c = a - 2\]
\[d = a - 3\]
Hence, we have \[{a^3} = {\left( {a - 1} \right)^3} + {\left( {a - 2} \right)^3} + {\left( {a - 3} \right)^3}\]
Now, \[a\] cannot be smaller than \[4\] because if it is, then \[d = a - 3\] is going to be less than \[1\] but a natural number cannot be less than \[1\].
Hence, we have established the fact that \[a \ge 4\].
So, let us put in \[a = 4\] and check for the question,
\[\begin{array}{l}LHS = {4^3}\\{\rm{ }} = 64\end{array}\]
\[\begin{array}{l}RHS = {\left( {4 - 1} \right)^3} + {\left( {4 - 2} \right)^3} + {\left( {4 - 3} \right)^3}\\{\rm{ }} = {\left( 3 \right)^3} + {\left( 2 \right)^3} + {\left( 1 \right)^3}\\{\rm{ }} = 27 + 8 + 1\\{\rm{ }} = 36 \ne LHS\end{array}\]
Now, let us try for \[a = 5\],
\[\begin{array}{l}LHS = {5^3}\\{\rm{ }} = 125\end{array}\]
\[\begin{array}{l}RHS = {4^3} + {3^3} + {2^3}\\{\rm{ }} = 64 + 27 + 8\\{\rm{ }} = 99 \ne LHS\end{array}\]
Now, let us try for \[a = 6\],
\[\begin{array}{l}LHS = {6^3}\\{\rm{ }} = 216\end{array}\]
\[\begin{array}{l}RHS = {5^3} + {4^3} + {3^3}\\{\rm{ }} = 125 + 64 + 27\\{\rm{ }} = 216 = LHS\end{array}\]
Hence, the minimum value of \[a = 6\].
Thus, the correct option is A.

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contain the known and the unknown and pick the one which is the most suitable and the most effective for finding the answer of the given question. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order, because even a slightest error is going to make the whole expression awry and is going to give us an incorrect answer.