question on remainder theorem................

Asked by krithika.r | 20th May, 2009, 03:38: PM

Expert Answer:

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

Answered by  | 2nd Sep, 2009, 02:53: PM

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