Question

If $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right),$ then x divides by 70 leaves a remainder of.
A.0
B.1
C.69
D.95

Answer 1

Hint: It is given in the question that if $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right)$ ,then x divides by 70 leaves a remainder of.
Since, we know that $\left( {a + b} \right)$ is a factor of $\left( {{a^3} + {b^3}} \right)$
Firstly, we apply the above condition on $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right)$ .
Then after, solving further we will get the required answer.

Complete step-by-step answer:
It is given in the question that if $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right)$ ,then x divides by 70 leaves a remainder of.
Since, $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right)$
We know that $\left( {a + b} \right)$ is a factor of $\left( {{a^3} + {b^3}} \right)$
Now, take 16 and 19 together and 17 and 18 together because 16 + 19 is 35 and 17 + 18 is 35.
Then, $\left( {{{16}^3} + {{19}^3}} \right)$ is divisible by 16+19=35 and
  $\left( {{{17}^3} + {{18}^3}} \right)$ is divisible by 17+18=35.
Therefore, it has a factor 35.
Also, $x = \left( {{{16}^3} + {{17}^3} + {{18}^3} + {{19}^3}} \right)$ is an even number.
Since, ${16^3} = 4096$ , ${17^3} = 4913$ , ${18^3} = 5832$ and ${19^3} = 6859$ .
  unit digit of ${16^3} + {17^3} + {18^3} + {19^3} = 6 + 3 + 2 + 9$
Therefore, unit digit =6+3+2+9=0. So, x has a factor 0.
Since, we know that E+O+E+O=E where E is even and O is odd.
So, x is completely divisible by $35 \times 2 = 70$ leaves a remainder 0.
Hence answer is option A.

Note: Remainder Theorem: In algebra, the polynomial remainder theorem is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial.
Factor Theorem: Factor theorem is generally applied to factorize and find the roots of polynomial equations. It is the reverse form of the remainder theorem. Problems are solved based on the application of synthetic division and then to check for a zero remainder.