Finding the Smallest Perfect Square Divisible by 6 and 12

The goal is to find the smallest number that is both a perfect square and divisible by both 6 and 12.

Calculate the Least Common Multiple (LCM)

First, find the LCM of 6 and 12. The LCM is the smallest number divisible by both numbers.

• Prime factorization of 6: $2 \times 3$
• Prime factorization of 12: $2^2 \times 3$
• LCM(6, 12) = $2^2 \times 3 = 12$.

So, 12 is the smallest number divisible by both 6 and 12.

Identify if LCM is a Perfect Square

Next, check if the LCM (12) is a perfect square. A perfect square has prime factors with even exponents.

• The prime factorization of 12 is $2^2 \times 3^1$.
• The exponent of 3 is odd (1).
• Therefore, 12 is not a perfect square.

Determine the Smallest Perfect Square Multiple

To find the smallest perfect square divisible by 12, we need to adjust the prime factorization of 12 ($2^2 \times 3^1$) so all exponents are even. We need to multiply by $3^1$ (which is 3) to make the exponent of 3 even.

• Smallest perfect square = LCM $\times$ (missing factors for even powers)
• Smallest perfect square = $(2^2 \times 3^1) \times 3^1 = 2^2 \times 3^2$
• Smallest perfect square = $4 \times 9 = 36$.

The number 36 is a perfect square ($6^2$) and is divisible by both 6 ($36 \div 6 = 6$) and 12 ($36 \div 12 = 3$).

Conclusion

The smallest perfect square number divisible by each of 6 and 12 is 36.