Question

If x²⁰²⁰ + 2019 is divided by (x+1) then the remainder​

Answer 1

Correct Question :- If x²⁰²⁰ is divided by (x+1) then the remainder is ?

concept used :-

• x^(Even Number) / (x + 1) always gives Remainder 1.

Solution :-

we have to Find the remainder when x²⁰²⁰ is divided by (x + 1) .

Checking the Power of x first, if its even or not.

we know that, Any number divisible by 2, is an Even number.

So,

→ 2020 ÷ 2 = 1010

Power of x is completely divisible by 2 . So, we conclude that, Power is an Even Number.

Therefore,

→ x²⁰²⁰ / (x + 1)

→ x^(Even Number) / (x + 1)

→ 1 Remainder. (Ans.)

Hence, The remainder will be 1.

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if in Question it is asked If (x²⁰²⁰ + 2019) is divided by (x+1) , where x is a constant number , then the remainder is ?

Than,

Let ,

→ p(x) = (x²⁰²⁰ + 2019)

we know that, when we divide p(x) by (x+1), we get the remainder p(-1). (a = bq + r, where r < b.)

Therefore,

On putting x = (-1) in p(x) , we get

→ p(x) = (x²⁰²⁰ + 2019)

→ p(-1) = (-1)²⁰²⁰ + 2019

→ p(-1) = 1 + 2019

→ p(-1) = 2020 (Ans.)

Hence, the remainder is 2020.

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Some Important Results to Remember :-

• x^(Even) / (x - 1) = Remainder 1.
• x^(Even) / (x + 1) = Remainder 1.
• x^(odd) / (x - 1) = Remainder 1.
• x^(odd) / (x + 1) = Remainder 1.
• (a^n - b^n) is divisible by (a + b)(a - b) when n is Even.
• (a^n - b^n) is divisible by (a - b) only when n is odd.