Determine the positive integer values of x less than 100 such that the last two digits of 2^x equal x.

What positive integer values of y less than 100 are there such that the last two digits of 3^y equal y?

For both 2^x and 3^y, the last two digits cycle every 20 numbers (except where x=1 for 2^x -- if represented as two digits; for this number, the last two digits do not equal to the last two digits of any number 2^x+20k, where k is a positive integer).

For the two digits to be equal to the exponent, the two digits must be true for the equation: (20*k - p) - d = 0, where d is the two digit number, k is a positive integer [between 0 and 5], and p is the position of the two digit number in the 20 number cycle. 

The only solutions are....for 2^x, x is 36; and, for 3^y, y is 87.

Posted by Dej Mar on 2006-08-09 03:20:51