What is the Least Square Number Divisible by 8, 9, 10?

The least square number divisible by 8, 9, and 10 is 3600.

To determine this, we need to find the least common multiple (LCM) of 8, 9, and 10 and then make sure the prime factorization of the LCM has even exponents for all prime factors. This will ensure it's a perfect square.

1. Prime Factorization:

   • 8 = 2³
   • 9 = 3²
   • 10 = 2 * 5

2. Least Common Multiple (LCM):
   To find the LCM, we take the highest power of each prime factor present in the factorizations:
   LCM(8, 9, 10) = 2³ 3² 5 = 8 9 5 = 360

3. Making it a Perfect Square:
   Now we analyze the exponents in the prime factorization of the LCM (360 = 2³ 3² 5¹). For a number to be a perfect square, all the exponents in its prime factorization must be even.

   • The exponent of 2 is 3 (odd). We need to multiply by 2 to make it 2⁴.
   • The exponent of 3 is 2 (even).
   • The exponent of 5 is 1 (odd). We need to multiply by 5 to make it 5².

   Therefore, we need to multiply the LCM by 2 * 5 = 10.

4. The Least Square Number:
   360 * 10 = 3600

   The prime factorization of 3600 is 2⁴ 3² 5². Since all the exponents are even, 3600 is a perfect square (60² = 3600) and is divisible by 8, 9, and 10.