What is the largest perfect square factor of 84
The Correct Answer and Explanation is :
To find the largest perfect square factor of 84, follow these steps:
Step 1: Prime Factorization
We start by finding the prime factorization of 84:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
So, the prime factorization of 84 is:
[ 84 = 2^2 \times 3 \times 7 ]
Step 2: Identify the Perfect Square Factors
A perfect square is a number that can be expressed as the square of an integer. When looking for perfect square factors, we want to find the largest factor whose prime factorization consists only of even powers of primes.
From the prime factorization of 84, we see:
- ( 2^2 ) is a perfect square (since it equals 4).
- ( 3 ) and ( 7 ) are not perfect squares by themselves, as their powers in the factorization are 1 (odd).
Thus, the only perfect square factor we can extract is ( 2^2 = 4 ).
Step 3: Conclusion
The largest perfect square factor of 84 is 4.
Explanation
The prime factorization of a number is key to understanding its divisibility properties, especially when looking for perfect square factors. A perfect square is any number that has even powers for all the primes in its factorization, as squaring a number doubles the exponent of its prime factors.
For 84, the only prime that contributes an even exponent is 2 (with an exponent of 2). This allows us to identify ( 2^2 = 4 ) as the largest perfect square factor. Since both 3 and 7 have odd exponents in the prime factorization of 84, they cannot contribute to a perfect square factor. Hence, 4 is the largest perfect square divisor of 84.
This method works for any number by breaking it down into prime factors and then checking for even exponents to identify the largest perfect square factor.