Shortcut Trick
Identify the prime factors of B = 201 × 202 × 203 × 204.
The factors of 201, 202, 203, and 204 are 3, 67, 2, 101, 7, 29, 4, 17 etc., which are all less than or equal to 200.
Since all prime factors of B are less than or equal to 200, they are already present in A = 200! with equal or higher powers.
Therefore, B completely divides A, which means the LCM of A and B is A.
∴ The correct answer is A.
Alternate Method
Given:
⇒ A = 200! = 1 × 2 × 3 × ... × 200
⇒ B = 201 × 202 × 203 × 204
Let us prime factorize each term of B:
⇒ 201 = 3 × 67
⇒ 202 = 2 × 101
⇒ 203 = 7 × 29
⇒ 204 = 4 × 3 × 17 = 22 × 3 × 17
The prime numbers involved in the factorization of B are 2, 3, 7, 17, 29, 67, 101.
Now, let us find the highest power of these prime numbers in A = 200! using Legendre's formula:
⇒ For prime 101: [200/101] = 1. Thus, 1011 divides A.
⇒ For prime 67: [200/67] = 2. Thus, 672 divides A.
⇒ For prime 29: [200/29] + [200/841] = 6. Thus, 296 divides A.
Since the highest exponent of each prime factor in A is greater than or equal to its exponent in B, B is a factor of A (i.e., B divides A).
⇒ LCM(A, B) = A
∴ The correct answer is A.
Additional Information
LCM of Divisible Numbers
If a number y completely divides x, then the Least Common Multiple LCM(x, y) = x and the Greatest Common Divisor GCD(x, y) = y.
Legendre's Formula
The exponent of a prime p in n! is given by Ep(n!) = [n/p] + [n/p2] + [n/p3] + ..., where [x] denotes the greatest integer function.