Algebra-20(CAT-2005)

If x = (16³ + 17³ + 18³ + 19³), then x divided by 70 leaves a remainder of

(1)0   (2) 1    (3) 69   (4) 35   (5) none of these

Solution follows here:

Solution:

x = 16³ + 17³ + 18³ + 19³

For such type of problems look for a small logic like, 16+19 = 17+18

x = (16³ + 19³) + (17³ + 18³)

Applying the formula, a³+b³ = (a+b)(a²-ab+b²):

=> x = (16+19)(16²-16.19+19²) + (17+18) (17²-17.19+19²)

=> x = (35)(16²-16.19+19²) + (35) (17²-17.19+19²)

From now onwards, no need of finding the values of these squares and multiplications. Just go on finding whether the resulting number is odd or even.

The logic here is:

odd+odd results in even; odd+even results in odd; even+even results in even; odd*odd results in odd; even*even results in even; odd*even results in even;

=> x = (35)(even-even+odd) + (35) (odd-odd+odd)

=> x = (35)(even+odd) + (35) (odd) => x = (35)(odd) + (35) (odd)

=> x = (35)(odd+odd)  => x = (35)(even)  => x = (35)[2*(some integer)]

=> x = (70)[some integer]  => ‘x’ is a multiple of 70

=> ‘x’ when divided by 70 leaves a remainder ‘0’.

Answer (1)