Let X and Y be the set of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then n(X Y) is equal to

Step-by-step solution

The prime factorization of 400 is 400 = 4 100 = 2^2 2^2 5^2 = 2^4 5^2 . The prime factorization of 1000 is 1000 = 10^3 = 2^3 5^3 . The set X contains all divisors of 2^4 5^2 , which are of the form 2^a 5^b where 0 a 4 and 0 b 2 . The set Y contains all divisors of 2^3 5^3 , which are of the form 2^c 5^d where 0 c 3 and 0 d 3 . The intersection X Y consists of all divisors that divide both 400 and 1000. These are the divisors of (400, 1000) . Calculating the greatest common divisor: (2^4 5^2, 2^3 5^3) = 2^ (4,3) 5^