Question

The number of distinct prime divisors of the number ${\left( {512} \right)^3} - {\left( {253} \right)^3} - {\left( {259} \right)^3}$ is
$
  (a){\text{ 4}} \\
  (b){\text{ 5}} \\
  (c){\text{ 6}} \\
  (d){\text{ 7}} \\
 $

Answer 1

Hint: In this question write 512 as a sum of 253+259. Then it gets in the form an algebraic identity${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$. To this simplified expression now find the number of distinct prime divisors.

Complete step-by-step answer:

Given equation is
${\left( {512} \right)^3} - {\left( {253} \right)^3} - {\left( {259} \right)^3}$
As we know (512 = 253 + 259) so use this we can written above equation as
$ \Rightarrow {\left( {253 + 259} \right)^3} - {\left( {253} \right)^3} - {\left( {259} \right)^3}$
Now as we know ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$ so use this property expand the cube in above equation we have,
$ \Rightarrow {\left( {253} \right)^3} + {\left( {259} \right)^3} + 3{\left( {259} \right)^2}\left( {253} \right) + 3\left( {259} \right){\left( {253} \right)^2} - {\left( {253} \right)^3} - {\left( {259} \right)^3}$
Now cancel out the terms we have,
$ \Rightarrow 3{\left( {259} \right)^2}\left( {253} \right) + 3\left( {259} \right){\left( {253} \right)^2}$
Now take common terms as common we have,
$ \Rightarrow 3\left( {259} \right)\left( {253} \right)\left( {259 + 253} \right)$
$ \Rightarrow 3\left( {259} \right)\left( {253} \right)\left( {512} \right)$
Now factorize the numbers we have,
$ \Rightarrow 3\left( {7 \times 37} \right)\left( {11 \times 23} \right)\left( {{2^9}} \right)$
$ \Rightarrow {2^9} \times 3 \times 7 \times 11 \times 23 \times 37$
Now as we know prime numbers are the numbers which can divide only by 1 or itself.
So in above factors of the given equation the set of prime factors are (2, 3, 7, 11, 23 and 37).
So the total number of distinct prime divisors of the number ${\left( {512} \right)^3} - {\left( {253} \right)^3} - {\left( {259} \right)^3}$ is 6.
Hence option (C) is correct.

Note: Prime numbers are those which are divisible by one and itself only, so prime divisors are the numbers which divide the given numbers and are prime as well. Direct evaluation of the given expression without simplifying can eventually be another method to solve this problem but it's very time consuming and lengthy.