lime777 wrote:How many pairs of positive integers x and y satisfy the equation x^y = 2013^2013?
2013 = 3*11*61
Hence, 2013^2013 = (3*11*61)^(3*11*61) = [(3*11*61)^3]^(11*61) = [(3*11*61)^11]^(3*61) = ... = [(3*11*61)^(3*11)]^61 = ... etc
Each of these above expression will give one pair of x and y. For example, for [(3*11*61)^11]^(3*61), x = [(3*11*61)^11] and y = (3*61)
Hence, number of pairs of x and y = Number of ways to select 1 or 2 or 3 integers among 3, 11, and 61 = (Number of ways to select 1 integer among 3 + Number of ways to select 2 integers among 3 + Number of ways to select 3 integers among 3) = 3
C1 + 3
C2 + 3
C3 = 3 + 6 + 1 = 10
lime777 wrote:How many pairs of positive integers x and y satisfy the equation x^y = 2012^2012?
2012 = (2^2)*503
The situation similar to the previous one, but now we have two identical factors. Hence, number of pairs of x and y will be exactly half of the previous one, i.e. 10/2 = 5
