Home » CAT Questions » Remainder when divided by 70 – CAT 2005 Question

Remainder when divided by 70 – CAT 2005 Question

Q.)If X = ({ 16 }^{ 3 }+{ 17 }^{ 3 }+{ 18 }^{ 3 }+{ 19 }^{ 3 }) , then X divided by 70 leaves a remainder of
(a)0
(b)1
(c)69
(d)35

[CAT 1995 Question , CAT Question, Number System , Remainder , Odd , Even , MBA Questions]

Solution :

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

{ 16 }^{ 3 } = even

{ 17 }^{ 3 } = odd

{ 18 }^{ 3 } = even

{ 19 }^{ 3 } = odd

even + odd + even + odd = even + even + odd + odd = even

Hence, X is divisible by 2 as X is even.

Now , We know that ({ a }^{ 3 }+{ b }^{ 3 }) = (a+b)({a}^{2} - ab + {b}^{2})

({ 16 }^{ 3 }+{ 17 }^{ 3 }+{ 18 }^{ 3 }+{ 19 }^{ 3 }) can be regrouped as ({ 16 }^{ 3 }+{ 19 }^{ 3 }+{ 17 }^{ 3 }+{ 18 }^{ 3 })

Now, ({ 16 }^{ 3 }+{ 19 }^{ 3 }) = (16+19)({16}^{2}-16*19+{19}^{2}) = 35({16}^{2}-16*19+{19}^{2})

({ 17 }^{ 3 }+{ 18 }^{ 3 }) = (17+18)({17}^{2}-17*18+{18}^{2}) = 35({17}^{2}-17*18+{18}^{2})

So, X will also be divisible by 35.

Hence, X is divisible by 2 and 35 both. Hence, X is divisible by 70.

So, the remainder obtained when X is divided by 70 = 0

Hence, OPTION(I)


Leave a comment