COMEDK 2023 Shift 1 Paper

To find n(X∩Y), the number of elements in the intersection of X and Y, we first need to determine the sets X and Y themselves. These correspond to the sets of all positive divisors of 400 and 1000 , respectively.
First, let's factorize 400 and 1000 :
‌400=24⋅52
‌1000=23⋅53
The set of divisors for a number n=pa⋅qb is given by varying the powers of p and q from 0 to their maximum in the factorization. Here, to find X∩Y, we need the maximum powers of primes that occur in both 400 and 1000 .
The greatest common divisor (GCD) of 400 and 1000 incorporates the lowest powers of the common prime factors:
For prime factor 2: the lower of 4 (from 400) and 3 (from 1000) is 3.
For prime factor 5 : the lower of 2 (from 400) and 3 (from 1000) is 2.
Thus, GCD(400,1000)=23⋅52=200.
The set of divisors of 200 (which represents X∩Y ) is formed by taking all combinations of 20,21,22,23 and 50,51,52 :
Divisors of 200:
‌20⋅50=1
‌20⋅51=5
‌20⋅52=25
‌21⋅50=2
‌21⋅51=10
‌21⋅52=50
‌22⋅50=4
‌22⋅51=20
‌22⋅52=100
‌23⋅50=8
‌23⋅51=40
‌23⋅52=200
Counting these, there are 12 divisors. Therefore, n(X∩Y)=12.
This corresponds to Option A: 12 .