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

To find the smallest square number divisible by 10, 16, and 24, we follow a two-step process: find the Least Common Multiple (LCM) and then ensure all prime factors have even exponents to form a perfect square.

Step-by-Step Calculation
1. Find the LCM of 10, 16, and 24:

Prime factorization:

10 = 2 * 5

16 = 2^4

24 = 2^3 * 3

Take the highest power of each prime factor present:

Factors: 2^4, 3^1, 5^1

LCM = 16 * 3 * 5 = 240.

2. Make the LCM a perfect square:
A perfect square must have all prime factors raised to an even power.

Prime factors of 240 = 2^4 * 3^1 * 5^1.

To make the exponents even:

2^4 is already even.

3^1 needs another 3 to become 3^2.

5^1 needs another 5 to become 5^2.

Multiply 240 by (3 * 5) = 15.

Smallest square number = 240 * 15 = 3600.