Question

If a, b, c are prime numbers such that \[{{a}^{3}}-{{b}^{3}}=c\] then which of the following is correct.
\[\begin{align}
  & 1.\text{ }a+b+c=24 \\
 & 2.\text{ }a+b+c=10 \\
 & 3.\text{ }a-b=5 \\
 & 4.\text{ }a+b=7 \\
\end{align}\]

Answer 1

Hint: Given a, b, c are prime numbers such that \[{{a}^{3}}-{{b}^{3}}=c\], then we have to check the correct option, which is satisfies the given expression. Firstly, we discuss the prime numbers. And then, we are going to substitute the prime numbers in the given expression and the options, if we get the same answer, we can easily find out the correct option.

Complete step-by-step solution:
Prime number definition: prime numbers are numbers that are divisible by itself and 1 only or the numbers whose only factors are the number itself and 1. Or A number is a factor of another number if it can divide it perfectly without any remainder.
Examples are \[2,3,5,7,11,13,17\]etc.
Let us solve the given question,
Given a, b, c are prime numbers
Such that \[{{a}^{3}}-{{b}^{3}}=c\]
Here, we are going to substitute any two prime numbers in the expression \[{{a}^{3}}-{{b}^{3}}=c\].
Let us consider\[a=3\], \[b=2\]
\[\Rightarrow {{a}^{3}}-{{b}^{3}}={{\left( 3 \right)}^{3}}-{{\left( 2 \right)}^{3}}\]
\[\Rightarrow {{a}^{3}}-{{b}^{3}}=27-8=19\]
\[\therefore c=19\].
Therefore, 19 is also a prime number.
Let us check with the options
\[1.a+b+c=2+3+19=24\]
\[3.a-b=3-2=1\]
\[4.a+b=3+2=5\]
Here, we are substituted the a, b, c values in the options,
Hence, we can conclude that only option (1) is satisfied.
Therefore, if a, b, c are prime numbers such that \[{{a}^{3}}-{{b}^{3}}=c\] then \[a+b+c=24\].

Note: We should be careful while doing these kinds of questions, we need to know what are the prime factors. So that we can easily substitute and solve the given expression accurately in a shorter amount of time. Understanding the concept is also a major concern.