what is the least number by which 198 should be divided so that the quotient is a perfect square
The Correct Answer and Explanation is:
To find the least number by which 198 should be divided so that the quotient is a perfect square, we begin by analyzing its prime factorization.
Step 1: Prime Factorization of 198
We divide 198 by its smallest prime factors:
198 ÷ 2 = 99
99 ÷ 3 = 33
33 ÷ 3 = 11
So, the prime factorization of 198 is:
198 = 2 × 3 × 3 × 11 = 2 × 3² × 11
Step 2: Understanding Perfect Squares
A perfect square is a number whose prime factors all occur in even powers. This means each prime number in the factorization must appear an even number of times.
In the factorization of 198:
- 2 occurs once (odd power)
- 3 occurs twice (even power)
- 11 occurs once (odd power)
To make the product a perfect square, we must eliminate or balance the odd powers of the primes. The primes with odd exponents are 2 and 11. To cancel out these odd powers and create even exponents, we should divide 198 by 2 × 11 = 22.
198 ÷ 22 = 9
Now check whether 9 is a perfect square:
Yes, 9 = 3², which is a perfect square.
Final Answer: 22
Explanation
The least number by which 198 should be divided to result in a perfect square is 22. This conclusion is based on the principle that perfect squares must have even powers in their prime factorizations. Since 198 has two primes (2 and 11) with odd powers, dividing by 22 cancels out those primes and leaves us with a number whose prime factors are all raised to even powers. The resulting quotient, 9, is a perfect square. Any number smaller than 22 would not fully eliminate the imbalance in the exponents and would not yield a perfect square. Therefore, 22 is the smallest such number.
