Let's determine whether 136 is a perfect square through a step-by-step analysis:
A perfect square is defined as a number that can be expressed as the square of an integer. In other words, if there exists an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 136 \)[/tex], then 136 would be a perfect square.
Here are the statements and a detailed examination of their correctness:
1. Yes, because [tex]\( 4 \cdot 34 = 136 \)[/tex], and 4 is a perfect square, so 136 is also a perfect square.
- This reasoning is incorrect. Although 4 is a perfect square, multiplying a perfect square (4) with another integer (34) does not necessarily result in a perfect square. For example, while 4 is a perfect square and 25 is also a perfect square, their product (4 * 25 = 100) is indeed a perfect square, but there is no general rule that the product of a perfect square and any integer is always a perfect square. Hence, 136 is not necessarily a perfect square just because 4 is a factor.
2. Yes, because [tex]\( 136 = 100 + 36 \)[/tex], and 100 and 36 are both perfect squares, so 136 will also be a perfect square.
- This reasoning is also incorrect. The sum of two perfect squares is not necessarily a perfect square. For example, the sum of 1 (which is [tex]\( 1^2 \)[/tex]) and 4 (which is [tex]\( 2^2 \)[/tex]) gives 5, which is not a perfect square. Therefore, even though 100 and 36 are perfect squares, their sum (136) is not necessarily a perfect square.
3. Yes, because perfect squares must be even numbers and 136 is an even number.
- This statement is incorrect. Being an even number does not imply being a perfect square. For example, 18 is an even number but not a perfect square. Therefore, 136 being an even number does not mean it is a perfect square.
4. No, because there is no whole number that when multiplied by itself gives 136.
- This reasoning is correct. To determine if 136 is a perfect square, we can check for a whole number whose square equals 136. The closest whole numbers are 11 and 12, where [tex]\( 11^2 = 121 \)[/tex] and [tex]\( 12^2 = 144 \)[/tex]. Since 136 lies between 121 and 144, it cannot be expressed as the square of any whole number.
Consequently, based on the correct analysis, the statement
"No, because there is no whole number that when multiplied by itself gives 136" is accurate.
Thus, 136 is not a perfect square.
