Maths

9997n

= 10000n - 3n

n must be odd.

So 10000n = ??.....A0000 , (here ??.....A = n, and A must be odd since n is odd)

Consider ??.....A0000 - K ,

If K between 1 to 10000, then ??......A0000 - K = ??....(A-1) ????.

So the million digit = A-1 is even ,hence K > 10000 for all digits of 9997n are odd.

So 3n > 10000 , n > 3333.333 , and n must be odd ,

So n is a odd number >= 3335,

when n = 3335 , 9997 * 3335 = 33339995

The smallest value of n is 3335


2009-12-09 17:29:14 補充：
Line 7 :

It should be Kilobit = A - 1 .

2009-12-09 17:29:35 補充：
Line 7 :

It should be Kilobit = A - 1 .

Since all digits are odd in 9997,
we can use the characteristics of numbers formed by 1 and 0 to simply repeat
all the digits again,
11 -> make a 1-digit number repeat ( 2 x 11 = 22)
101-> make a 2-digit number repeat ( 22 x 101 = 2,222)
1001 -> make a 3-digit number repeat ( 222 x 1001 = 222,222)
10001 -> make a 4-digit number repeat ( 2222 x 10001 = 22,222,222)

9997 x 10001 = 99979997.
All numbers are odd.

There can't have any smaller number, since repeating the digit is the only way.

As 7 x an even number can only be an even number, all even number is not
suitable.

As all odd numbers time odd number can only be odd numbers, the addition of
digits(進位) must be even number in order to make the number odd, but the
hundreds digit if the number will be even too.

So addition of digits(進位) must not exists in order to make the number odd.

As odd number plus any odd number can only results a even number,no
addition of odd number digits and odd number digits is allowed in the period of multiplication

The digits which follows these criteria are only 1, 10001, 11111 ...( The numbers which the digits are only made of 1 and 0 in the middle )

As the wuestion requires, the number must be greater than 1, which means cannot be 1, so the smallest number is 10001.

2009-12-07 19:58:58 補充：
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「'」 係 「'」 既符號黎.