To find the third number given the highest common factor (HCF) and least common multiple (LCM) of 3240, 3600, and another number, we first need to recap how HCF and LCM work.
1. **Understanding HCF and LCM**: The HCF of two or more numbers is the largest number that divides them without leaving a remainder. The LCM is the smallest number that is a multiple of both numbers. In this case, we need the HCF and LCM of the three numbers.
2. **Finding the LCM**: The prime factorization of the numbers can help us here.
- **3240**: Its prime factors are 23 × 34 × 51.
- **3600**: Its prime factors are 24 × 32 × 52.
3. **Finding the HCF**: To find the HCF of these two numbers, we take the lowest power of each prime factor present in both numbers.
- For 2, the lowest power is 23.
- For 3, the lowest power is 32.
- For 5, the lowest power is 51.
Thus, HCF(3240, 3600) = 23 × 32 × 51 = 120.
4. **Finding the relationship**: Given that the LCM of all three numbers is already provided as 2, 4, 3, 5, 5, 2, 7, 2 (which can be factored accordingly). After decomposing it into its prime factors, we need to find how they relate to the numbers.
5. **Finding the third number**: With the relationships of HCF(3240, 3600, x) = 36, we can derive that ‘x’ would need to be aligned in the prime factorization ratios. We have to ensure that:
- HCF(3240, 3600, x) = 36
- LCM(3240, 3600, x) matches the prime multiplication to 2 × 2 × 3 × 5 × 5 × 2 × 7 × 2.
Through adjustment, we find that if the prime factors of x align correctly with LCM conditions, we derive the value of the third number as 180.
In conclusion, the third number that completes the equation with 3240, 3600, and their LCM as stated is 180.