Prime factorization of 20 in exponential form is: 22x5
They are: 10*11 = 110
You can multiply 2 times 11 times 5 to make 110! (2x11=22x5=110)
8=23 12=22x3 20=22x5 28=22x7 18=2x32 27=33
4=22 10=2x5 So the LCM of 4 and 10 is 22x5=20 and the next multiples of both 4 and 10 are multiples of 20
If a number is a perfect square, then its prime factors can be split into two identical groups: if a perfect cube, three groups; if a perfect fifth power, five groups. The best way to attain this sort of 'symmetry' is to only use one prime factor. Since we are looking for small (ish) solutions, we'll use 2. It only remains to decide how many 2's to multiply together. The number needs to be divisible by 2, 3 and 5. We are looking for the lowest common denominator of 2, 3 and 5, which is 30. 22x3x5 = 230 = 1073741824 = (22x3)5 = (22x5)3 = (23x5)2 The next smallest solution is given by 330 = 205891132094649, which is just a little bit bigger than 2 x 1014. So the question is correct in assuming that only one answer exists.