CBSE Class 10 Answered

aⁿ + bⁿ is divisible by a+b when n is odd.

and,  aⁿ + bⁿ is divisible by a-b when n is even.

But to make this property applicable for the number 5²⁰⁰⁹ +3²⁰⁰⁹  the divisor should be 8. Please confirm whether its 8 or 18.

2. To find out the remainder when 5²⁰⁰⁹ +3²⁰⁰⁹ is divided by 18 we will use chinese remainder theorem.

We will individually compute the remainder for each term 5²⁰⁰⁹  and 3²⁰⁰⁹ respectively.

Case 1: Remainder when 3²⁰⁰⁹ is divided by 18.

3²⁰⁰⁹/ 18 = 3²⁰⁰⁷/ 2

⇒  3²⁰⁰⁷ = (1)²⁰⁰⁷ (mod2)

                 = 1 (mod2)

Remainder = 1

Case 2: Remainder when 5²⁰⁰⁹ is divided by 18.

In this case, we will divide 5²⁰⁰⁹ by the factors of 18 which are 2 and 9

5²⁰⁰⁹/ 2

⇒5²⁰⁰⁹ = (1)²⁰⁰⁹ (mod2)

                 = 1 (mod2)

Now, for 5²⁰⁰⁹/ 9

⇒ 5²⁰⁰⁹ = (5³)⁶⁶⁹.5²

                 =  (-1)⁶⁶⁹ .5² (mod9)

                =  -25 (mod9)

                = 2 (mod9)

Now from chinese remainder theorem, we know that,

For two simultaneous congruences

n = n₁ (mod m₁) and

n = n₂ (mod m₂) are onlly solvable when n₁ = n₂ (mod gcd(m₁, m₂))

Using the above theorem for our two simulataneous congreunces,

 5²⁰⁰⁹ = 2 (mod9) ................(A)

 5²⁰⁰⁹  = 1 (mod2) ............(B)

So, 2 = 1 (modgcd(9,2))

        2  = 1 (mod 1)

Therefore, 2 = t +1

and t = -1

Remainder when 5²⁰⁰⁹ +3²⁰⁰⁹ is divided by 18 is the sum of remainders when each term 5²⁰⁰⁹, 3²⁰⁰⁹ is divided by 18,

From (1) and (2)

Remainder = 1 + -1 = 0