Choose the correct answer.Written as a product of its prime factors, 1176 = 23 × 3 × 72. Find the smallest positive integer y such thatis a square number.5678

To find the smallest positive integer $y$ such that $1176 \times y$ is a square number, we need to determine the prime factorization of $1176$.

1. Break Down the Problem

   We start with the prime factorization provided: $1176 = 2^3 \times 3^1 \times 7^2$

2. Relevant Concepts

   For a number to be a perfect square, all exponents in its prime factorization must be even.

3. Analysis and Detail

   • The current prime factorization shows:
     • $2^3$: exponent is 3 (odd)
     • $3^1$: exponent is 1 (odd)
     • $7^2$: exponent is 2 (even)

   To make all the exponents even, we need to:

   • Increase the exponent of $2$ from 3 to 4 (add $1$, thus multiply by $2^1$)
   • Increase the exponent of $3$ from 1 to 2 (add $1$, thus multiply by $3^1$)
   • The exponent of $7$ is already even, so no need to multiply by $7^0$.

   Therefore, we multiply by: $y = 2^1 \times 3^1 = 6$

4. Verify and Summarize

   So, to make $1176 \times y$ a perfect square: $1176 \times 6 = 1176 \times (2^1 \times 3^1)$

   The resulting prime factorization will be $2^4 \times 3^2 \times 7^2$, where all exponents are even, confirming that it is indeed a perfect square.

Final Answer

The smallest positive integer $y$ is $\boxed{6}$.