Basics : Number System Remainders 1

When dividend is of the form a ⁿ + b ⁿ or a ⁿ - b ⁿ :

8. What is the remainder when (5555) ²²²² + (2222) ⁵⁵⁵⁵ is divided by 7?

       The remainders when 5555 and 2222 are divided by 7 are 4 and 3 respectively.

       Hence, the problem reduces to finding the remainder when (4) ²²²² + (3) ⁵⁵⁵⁵ is divided by 7.

       Now (4) ²²²² + (3) ⁵⁵⁵⁵ = (4 ² ) ¹¹¹¹ + (3 ⁵ ) ¹¹¹¹ = (16) ¹¹¹¹ + (243) ¹¹¹¹ .

        Now (16) ¹¹¹¹ + (243) ¹¹¹¹ is divisible by 16 + 243 or
        it is divisible by 259, which is a multiple of 7.

       Hence the remainder when (5555) ²²²² + (2222) ⁵⁵⁵⁵ is divided by 7 is zero.

9. 20 ²⁰⁰⁴ + 16 ²⁰⁰⁴ - 3 ²⁰⁰⁴ - 1 is divisible by:    (a) 317 (b) 323 (c) 253 (d) 91

         20 ²⁰⁰⁴ + 16 ²⁰⁰⁴ - 3 ²⁰⁰⁴ - 1 = (20 ²⁰⁰⁴ - 3 ²⁰⁰⁴ ) + (16 ²⁰⁰⁴ - 1 ²⁰⁰⁴ )

         20 ²⁰⁰⁴ - 3 ²⁰⁰⁴ is divisible by 17 (Theorem 3) and
        16 ²⁰⁰⁴ - 1 ²⁰⁰⁴ is divisible by 17 (Theorem 2).
        Hence the complete expression is divisible by 17.

        20 ²⁰⁰⁴ + 16 ²⁰⁰⁴ - 3 ²⁰⁰⁴ - 1 = (20 ²⁰⁰⁴ - 1 ²⁰⁰⁴ ) + (16 ²⁰⁰⁴ - 3 ²⁰⁰⁴ ).

        Now 20 ²⁰⁰⁴ - 1 ²⁰⁰⁴ is divisible by 19 (Theorem 3) and
       16 ²⁰⁰⁴ - 3 ²⁰⁰⁴ is divisible by 19 (Theorem 2).

       Hence the complete expression is also divisible by 19. is divisible by 17 × 19 = 323.

 Fermat's Theorem

12. What is the remainder when n ⁷ - n is divided by 42?

Since 7 is prime, n ⁷ - n is divisible by 7.
 n ⁷ - n = n(n ⁶ - 1) = n (n + 1)(n - 1)(n ⁴ + n ² + 1)

Now (n - 1)(n)(n + 1) is divisible by 3! = 6

Hence n ⁷ - n is divisible by 6 x 7 = 42. Hence the remainder is 0.

 Wilson's Theorem

13. Find the remainder when 16! Is divided by 17.

              16! = (16! + 1) -1 = (16! + 1) + 16 - 17

              Every term except 16 is divisible by 17 in the above expression.
              Hence the remainder = the remainder obtained when 16 is divided by 17 = 16