Finding the Least Perfect Square Factor

The question asks for the smallest number that is both a perfect square and has 3675 as a factor.

Prime Factorization of 3675

1. First, find the prime factorization of 3675.
   • $3675 = 3 \times 1225$
   • $1225 = 5 \times 245$
   • $245 = 5 \times 49$
   • $49 = 7 \times 7 = 7^2$
2. So, the prime factorization is $3^1 \times 5^2 \times 7^2$.

Constructing the Least Perfect Square

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

• The prime factorization of 3675 is $3^1 \times 5^2 \times 7^2$.
• The exponents for 5 ($2$) and 7 ($2$) are already even.
• The exponent for 3 is $1$, which is odd. To make it even, we need to increase it to the next even number, which is $2$.
• Therefore, the prime factorization of the least perfect square containing 3675 must be $3^2 \times 5^2 \times 7^2$.

Calculation

Calculate the value of the number with the required prime factorization:

• The number is $3^2 \times 5^2 \times 7^2$.
• This can be written as $(3 \times 5 \times 7)^2$.
• Calculate the base: $3 \times 5 \times 7 = 15 \times 7 = 105$.
• The number is $105^2$.
• $105^2 = 11025$.

Verification

The number 11025 is a perfect square ($105^2$) and contains 3675 as a factor ($11025 \div 3675 = 3$). Thus, 11025 is the least such number.

Final Answer

The least number which is a perfect square and contains 3675 as its factor is 11025.