What is the greatest perfect square that is a factor of 147?
The Correct Answer and Explanation is :
To find the greatest perfect square that is a factor of 147, we need to start by finding the prime factorization of 147.
Step 1: Prime Factorization of 147
Start by dividing 147 by the smallest prime, which is 2. Since 147 is an odd number, it is not divisible by 2.
Next, check for divisibility by 3:
- The sum of the digits of 147 is ( 1 + 4 + 7 = 12 ), which is divisible by 3.
- So, divide 147 by 3:
[
147 \div 3 = 49
]
Now, we have ( 147 = 3 \times 49 ).
Next, factor 49. Since 49 is ( 7^2 ), we have:
[
49 = 7 \times 7
]
Thus, the complete prime factorization of 147 is:
[
147 = 3 \times 7^2
]
Step 2: Identify the Perfect Square Factors
A perfect square is a number whose prime factors all appear with even exponents. To find the greatest perfect square factor of 147, we examine the prime factorization:
[
147 = 3^1 \times 7^2
]
- The factor ( 3^1 ) has an odd exponent, so it cannot contribute to a perfect square.
- The factor ( 7^2 ) has an even exponent, so it can contribute to a perfect square.
Thus, the greatest perfect square factor is ( 7^2 = 49 ).
Step 3: Verify
We now check if 49 is a factor of 147:
[
147 \div 49 = 3
]
Since 3 is an integer, 49 is indeed a factor of 147.
Conclusion
The greatest perfect square that is a factor of 147 is ( \boxed{49} ).