Finding the Smallest Square Number
To find the smallest square number that is divisible by 6, 9, and 15, we first need to determine the least common multiple (LCM) of these numbers.
Step 1: Prime Factorization
- 6 = 2 × 3
- 9 = 3²
- 15 = 3 × 5
Step 2: Determine the LCM
To find the LCM, take the highest power of each prime number present in the factorizations:
- 2: Highest power is 2¹ from 6.
- 3: Highest power is 3² from 9.
- 5: Highest power is 5¹ from 15.
Thus, the LCM is:
- LCM = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90
Step 3: Finding the Smallest Square Number
A square number must have even powers of all prime factors. Currently, in the LCM (90):
- 2: 1 (odd)
- 3: 2 (even)
- 5: 1 (odd)
To make the powers even for each prime:
- For 2: Multiply by 2 (to make it 2²).
- For 3: No change needed (already even).
- For 5: Multiply by 5 (to make it 5²).
Thus, we need to multiply the LCM (90) by 2 and 5:
- Required number = 90 × 2 × 5 = 900
Step 4: Conclusion
The smallest square number divisible by 6, 9, and 15 is:
- 900
This is because 900 = 30², confirming that it is a perfect square.