(1)Is there an integer N such that 1998*N = 22222.......22222 (only the digit 2 in the expression of this number)? If so, how many digits are in N?

(2)A pocket calculator is broken. It is only possible to use the function keys: + , - , =, 1/x(inverse function). All number keys and the memory funtion work. How can we calculate the product 37 * 54? (The result is obviously 1998.)

(3)Is it true that 11111^99999 + 99999^88888 is divisible by 1998? Same question with 111111^999999 + 999999^888888 ?

One way is to add 54 to itself 37 times.

We could probably also factor the numbers but that isn't in the spirit of the question as division is not allowed.

How about this: Look at 1/37 = .027027027... which is clearly 27/999
since 27 is half of 54, the product is equivalent to 999+999
(this sort of uses division though)

Posted by Jer on 2013-12-05 15:55:41