The largest perfect square number that is a factor of 75 is?
the largest perfect square number that is a factor of 32 is?
the largest perfect square number that is a factor of 68 is?
The Correct Answer and Explanation is :
To find the largest perfect square that is a factor of a given number, we first need to understand what a perfect square is. A perfect square is an integer that can be expressed as the square of another integer. For example, (1, 4, 9, 16, ) and (25) are all perfect squares.
1. Factors of 75
First, we find the prime factorization of 75:
[
75 = 3 \times 5^2
]
The factors of 75 are (1, 3, 5, 15, 25, 75). Among these, the perfect squares are (1) and (25). Therefore, the largest perfect square factor of 75 is:
[
\boxed{25}
]
2. Factors of 32
Next, we analyze 32:
[
32 = 2^5
]
The factors of 32 are (1, 2, 4, 8, 16, 32). The perfect squares in this list are (1) and (4). Consequently, the largest perfect square factor of 32 is:
[
\boxed{16}
]
3. Factors of 68
Now, let’s find the factors of 68:
[
68 = 2^2 \times 17
]
The factors of 68 are (1, 2, 4, 17, 34, 68). The perfect squares among these factors are (1) and (4). Thus, the largest perfect square factor of 68 is:
[
\boxed{4}
]
Conclusion
In summary, the largest perfect square factors of the numbers are as follows:
- For 75, the largest perfect square factor is (25).
- For 32, it is (16).
- For 68, it is (4).
This method involves identifying the prime factorization of each number and then determining the highest powers of the prime factors that can form perfect squares. Perfect squares can only be formed from even powers of prime factors, which is why we consider the largest even power available in the factorization.