Answer
Verified
412.2k+ views
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.
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.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE