HCF of 3240 and 3600 and a third number is 36 and their LCM is 24 × 35 × 52 × 72. Then the third number is?

The HCF (Highest Common Factor) of two numbers 'a' and 'b' is the greatest possible number that divides both 'a' and 'b' completely. 

Answer: The third number is 47628 (2² × 3⁵ × 7²)

Let us see how to solve it.

Explanation:

To solve this question, we first need to express the numbers in the form of prime factors.

3240 = 2³ × 3⁴ × 5

3600 = 2⁴ × 3² × 5²

The HCF is 36 which can be expressed as 2² × 3² by prime factorization.

The LCM in the form of prime factors is 2⁴ × 3⁵ × 5² × 7²

By the definition, we know that HCF is the product of the lowest power of common prime factors. This means that the third number must include the HCF, which is 2² × 3².

Also, LCM is the product of the highest power of common prime factors implying that the third number must include 2² × 7²

Thus, the third number is 2² × 3⁵ × 7².

Therefore, the third number is 47628 (2² × 3⁵ × 7²)