The least perfect square number which is exactly divisible by 3, 4, 7, 10 and…

To find the least perfect square number divisible by 3, 4, 7, 10, and 12, we must find their Least Common Multiple (LCM) and then ensure all prime factors have even exponents.

Step-by-Step Calculation
1. Find the LCM of 3, 4, 7, 10, and 12:

• Prime factorization:

• 3 = 3¹

• 4 = 2²

• 7 = 7¹

• 10 = 2¹ × 5¹

• 12 = 2² × 3¹

The LCM is the product of the highest power of each prime factor present:

• Factors: 2², 3¹, 5¹, 7¹

• LCM = 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7 = 420.

2. Make the LCM a perfect square:
A number is a perfect square only if every prime factor has an even exponent.

• Current factors of 420: 2² × 3¹ × 5¹ × 7¹

• To make exponents even, we need to multiply by missing factors to pair them up:

• 2² is already even.

• 3¹ needs a 3¹ to become 3².

• 5¹ needs a 5¹ to become 5².

• 7¹ needs a 7¹ to become 7².

• Missing factors to multiply: 3 × 5 × 7 = 105.

• Perfect square = 420 × 105 = 44100.