What is the smallest square number which is divisible by 4, 9 and 10?

The smallest square number divisible by 4, 9, and 10 is 900.

Here's how we find that:

1. Prime Factorization: Find the prime factorization of each number:

   • 4 = 2²
   • 9 = 3²
   • 10 = 2 x 5

2. Least Common Multiple (LCM): Find the LCM of 4, 9, and 10. To do this, take the highest power of each prime factor present in the factorizations:

   • LCM(4, 9, 10) = 2² x 3² x 5 = 4 x 9 x 5 = 180

3. Making it a Perfect Square: For a number to be a perfect square, all the exponents in its prime factorization must be even. The prime factorization of 180 is 2² x 3² x 5¹. We notice that the exponent of 5 is odd (1). To make it even, we need to multiply by 5.

4. Calculate the Square Number: Multiply the LCM by the necessary factor(s) to make all exponents even: 180 x 5 = 900.

5. Verification: Check that 900 is a perfect square: √900 = 30. Also, confirm that 900 is divisible by 4, 9, and 10:

   • 900 / 4 = 225
   • 900 / 9 = 100
   • 900 / 10 = 90

Since 900 is a perfect square and is divisible by 4, 9, and 10, it's the smallest such number.