Thursday, 3 November 2011

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’.