- What is the remainder when the product 1998 × 1999 × 2000 is divided by 7?
Answer: the remainders when 1998, 1999, and 2000 are divided by 7 are 3, 4, and 5 respectively. Hence the final remainder is the remainder when the product 3 × 4 × 5 = 60 is divided by 7.
Answer = 4
- What is the remainder when 2 2004 is divided by 7?
2 2004 is again a product (2 × 2 × 2... (2004 times)). Since 2 is a number less than 7 we try to convert the product into product of numbers higher than 7. Notice that 8 = 2 × 2 × 2. Therefore we convert the product in the following manner-
2 2004 = 8 668 = 8 × 8 × 8... (668 times).
The remainder when 8 is divided by 7 is 1.
Hence the remainder when 8 668 is divided by 7 is the remainder obtained when the product 1 × 1 × 1... is divided by 7
Answer = 1
- What is the remainder when 2 2006 is divided by 7?
This problem is like the previous one, except that 2006 is not an exact multiple of 3 so we cannot convert it completely into the form 8 x . We will write it in following manner-
2 2006 = 8 668 × 4.
Now, 8 668 gives the remainder 1 when divided by 7 as we have seen in the previous problem. And 4 gives a remainder of 4 only when divided by 7. Hence the remainder when 2 2006 is divided by 7 is the remainder when the product 1 × 4 is divided by 7.
Answer = 4
- What is the remainder when 25 25 is divided by 9?
Again 25 25 = (18 + 7) 25 = (18 + 7)(18 + 7)...25 times = 18K + 7 25
Hence remainder when 25 25 is divided by 9 is the remainder when 7 25 is divided by 9.
Now 7 25 = 7 3 × 7 3 × 7 3 .. (8 times) × 7 = 343 × 343 × 343... (8 times) × 7.
The remainder when 343 is divided by 9 is 1 and the remainder when 7 is divided by 9 is 7.
Hence the remainder when 7 25 is divided by 9 is the remainder we obtain when the product 1 × 1 × 1... (8 times) × 7 is divided by 9. The remainder is 7 in this case. Hence the remainder when 25 25 is divided by 9 is 7.
- What the remainder when 2 96 is divided by 96?
The common factor between 2 96 and 96 is 32 = 2 5 .
Removing 32 from the dividend and the divisor we get the numbers 2 91 and 3 respectively.
The remainder when 2 91 is divided by 3 is 2.
Hence the real remainder will be 2 multiplied by common factor 32.
Answer = 64
- Find the remainder when 7 52 is divided by 2402.
Answer: 7 52 = (7 4 ) 13 = (2401) 13 = (2402 - 1) 13 = 2402K + (-1) 13 = 2402K - 1.
Hence, the remainder when 7 52 is divided by 2402 is equal to -1 or 2402 - 1 = 2401.
Answer: 2401.
2 .1C When dividend is of the form a n + b n or a n - b n :
EXAMPLES
- What is the remainder when 3 444 + 4 333 is divided by 5?
Answer:
The dividend is in the form a x + b y . We need to change it into the form a n + b n .
3 444 + 4 333 = (3 4 ) 111 + (4 3 ) 111 .
Now (3 4 ) 111 + (4 3 ) 111 will be divisible by 3 4 + 4 3 = 81 + 64 = 145.
Since the number is divisible by 145 it will certainly be divisible by 5.
Hence, the remainder is 0.
- What is the remainder when (5555) 2222 + (2222) 5555 is divided by 7?
Answer:
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.
- 20 2004 + 16 2004 - 3 2004 - 1 is divisible by:
(a) 317 (b) 323 (c) 253 (d) 91
Solution: 20 2004 + 16 2004 - 3 2004 - 1 = (20 2004 - 3 2004 ) + (16 2004 - 1 2004 ).
Now 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 Hence the complete expression is divisible by 17 × 19 = 323.
EXAMPLE
- What is the remainder when n 7 - n is divided by 42?
Answer: 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.
2.1F Wilson's Theorem
EXAMPLE
- 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
Answer = 16
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