Question

If $x=({16}^3+{17}^3+{18}^3+{19}^3)$, then $x$ divided by $70$ leaves a reminder of? (CAT 2005)

• A. 1
• B. 69
• C. 35

Answer 1

Sol:

$x={16}^3+{17}^3+{18}^3+{19}^3$

$=({16}^3+{19}^3)+({17}^3+{18}^3)$

$=(16+19)({16}^2+16\times 19+{19}^2)+(17+18)({17}^2+17\times 18+{18}^2)$

$=35\times$ (an odd number) $+35\times$ (another odd number)

$=35\times$ (an even number) $=35\times \left(2k\right)..........$ (k is a positive integer)

$x=70k$

Therefore x is divisible by 70. Remainder when x is divided by 70 = 0

Hence, option 1.