Determine the total number of pairs (x, y) of positive integers such that the least common multiple (LCM) of x and y is 2^{3}*5^{7}*11^{13}.
For choosing the powersof2 components of the two numbers, each can have from 0 to 3 as its power of 2 (4 choices each) but they can't both have from 0 to 2 as each one's power of 2 (3 choices each). So the choices for power of 2 are (4^2  3^2), and similarly for the larger powers for the other primes listed:
(4*43*3)*(8*87*7)*(14*1413*13) = 2835
FOR x2 = 0 TO 3
FOR y2 = 0 TO 3
FOR x5 = 0 TO 7
FOR y5 = 0 TO 7
FOR x11 = 0 TO 13
FOR y11 = 0 TO 13
IF (x2 = 3 OR y2 = 3) AND (x5 = 7 OR y5 = 7) AND (x11 = 13 OR y11 = 13) THEN ct = ct + 1
NEXT
NEXT
NEXT
NEXT
NEXT
NEXT
PRINT ct
confirms the result.

Posted by Charlie
on 20131014 10:38:52 