SAT (Stage-1) Mock Test - 11

Let f(x) = x⁴ + 2x³ - 2x² + x - 1 is divided by x² + 2x - 3.

x⁴ + 2x³ - 2x² + x -1 = {(x² + 2x -3) (x² + 1)} - x + 2

i.e. we get remainder a= -x + 2.

Hence, x - 2 must be added to x⁴ + 2x³ - 2x² + x - 1, so that the result is exactly divisible by x² + 2x - 3.

i.e x - 2 should be added.

2⁴ = 16; 2⁸ = 256; 2⁴ⁿ ends with 6
16 - 1 = 15, divisible by 5
2⁶⁰ ends with the unit digit as 6.
(6 - 1) will end in 5.
So, 2⁶⁰ - 1 is divisible by 5.
Remainder = 0

A number is divisible by 30 only if it is divisible by 3 and 10, so for divisibility by 10, unit's place digit should be 0, so y = 0.

And for divisibility by 3, sum of numbers should be divisible by 3.

i.e. 3 + 6 + 5 + x + y = 14 + x is divisble by 3.

So, the least value of x in this case will be 1.

As 15 = 14 + 1 is divisible by 3,

So, least value of x + y = 1 + 0 = 1

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