Factorisations

Since $gcd(a,b) = 1$ the exponents are in fact irrelevant, all that matters is the sorting of primes between a and b . If there were no other conditions, this could be done in $2^7$ way since there are 7 primes and for each prime we can either put it in a or put it in b . However, for each solution $(x,y)$ there is also the solution $(y,x)$ , and, since x never equals y , in half of the solutions $a < b$ . Therefore the number of solutions where $a > b$ is $\dfrac{2^7}{2} = 2^6$ . However, this also includes the solution where $b = 1$ , so there are in fact $2^{6} - 1 = \fbox{63}$ solutions.

all are prime factor (i.e. 2,3,5,6,7,11,13,17) . now consider two urn let A and B . now first put any one factor(as mention earlier) in to A then obviously B contain rest of the six. this can be done by 7C1 ways. now put any two in A then B contain rest 5 .this can be done 7C2 way. now choose any three and this can be done 7C3 ways.. now we should not consider the rest of the case i.e if A contained 4,5,6 numbers.. because this are the repeating case of B's . so the total number of ways which satisfied gcd(a,b)=1 is as follows 7C1+7C2+7C3=63