The smallest square number divisible by 10,16 and 24 is

Smallest Square Number Divisible by 10, 16, and 24

Step 1: Find the LCM of 10, 16, and 24

1. Prime factorization of the numbers:

• 10 = 2 × 5
• 16 = 2⁴
• 24 = 2³ × 3

2. LCM calculation:

For the LCM, take the highest powers of all prime factors:

• The highest power of 2 is 2⁴.
• The highest power of 3 is 3¹.
• The highest power of 5 is 5¹.

So, the LCM of 10, 16, and 24 is:

LCM = 2⁴ × 3 × 5 = 16 × 3 × 5 = 240.

Step 2: Adjust LCM to be a perfect square

For a number to be a perfect square, all the powers of its prime factors must be even.

In the prime factorization of 240: 2⁴ × 3¹ × 5¹, the powers of 3 and 5 are odd. To make these even, we need to multiply by 3 and 5.

Thus, we need to multiply 240 by 3 × 5 = 15.

Step 3: Multiply to get the smallest square number

Smallest square number = 240 × 15 = 3600.

Final Answer: The smallest square number divisible by 10, 16, and 24 is 3600.