What is the least number which is a perfect square and contains 3675 as its factor?

Correct option is A

Given:

We need to find the least number which is a perfect square and contains 3675 as a factor.

Solution:

We start by factoring 3675:

3675 is divisible by 5 (since it ends in 5):

3675 ÷ 5 = 735

735 is also divisible by 5:

735 ÷ 5 = 147

147 is divisible by 3 (since the sum of digits 1 + 4 + 7 = 12, which is divisible by 3):

​$$147 \div 3 = 49$$​

49 is $$7^2.$$​​

Thus, the prime factorization of 3675 is:

3675 =$$5^2 \times 3 \times 7^2$$​

To make 3675 a perfect square, we need all the exponents in its prime factorization to be even. Currently:

​$$5^2$$ already has an even exponent.

​$$3^1$$ has an odd exponent.

​$$7^2$$ already has an even exponent.

We need to multiply by $$3^1$$​ (to make the exponent of 3 even) to make the number a perfect square.

Thus, the least number which is a perfect square and contains 3675 as a factor is:

​$$3675 \times 3 = 11025$$​

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