Question

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

• A. 12
• B. 10
• C. 8
• D. 6

Hint:

To find the common divisors of two numbers, first find the greatest common divisor (gcd), then list the divisors of the gc(D)

Answer 1

The Correct Option is A

The divisors of 400 are given by the prime factorization: \[ 400 = 2^4 \times 5^2 \] The number of divisors of 400 is given by \((4+1)(2+1) = 15\). The divisors are: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400. Similarly, the divisors of 1000 are: \[ 1000 = 2^3 \times 5^3 \] The number of divisors of 1000 is \((3+1)(3+1) = 16\). The divisors are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000. Now, to find the divisors common to both 400 and 1000, we calculate the greatest common divisor (gcd) of 400 and 1000. \[ \gcd(400, 1000) = 2^3 \times 5^2 = 200 \] The divisors of 200 (the gcd) are the common divisors of 400 and 1000. The divisors of 200 are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200. Thus, the number of common divisors, \( n(X \cap Y) \), is 12.