We can find the smallest number by which each given number must be multiplied to make it a perfect square by first factorizing each number into its prime factors, and then identifying the missing factors that would make it a perfect square.
Let's take each number one by one:
156 = 2² x 3 x 13. To make this a perfect square, we need to multiply it by 13, since that's the missing factor. So the smallest number by which 156 must be multiplied to make it a perfect square is 13.
1331 = 11³. This is already a perfect cube, and therefore a perfect square as well. So the smallest number by which 1331 must be multiplied to make it a perfect square is 1.
432 = 2⁴ x 3³. To make this a perfect square, we need to multiply it by 2, since that's the missing factor. So the smallest number by which 432 must be multiplied to make it a perfect square is 2.
700 = 2² x 5² x 7. To make this a perfect square, we need to multiply it by 5 x 7, since those are the missing factors. So the smallest number by which 700 must be multiplied to make it a perfect square is 5 x 7 = 35.
882 = 2 x 3² x 7². This is already a perfect square, since it has an even power of each prime factor. So the smallest number by which 882 must be multiplied to make it a perfect square is 1.
3698 = 2 x 7 x 13 x 19. To make this a perfect square, we need to multiply it by 2 x 19, since those are the missing factors. So the smallest number by which 3698 must be multiplied to make it a perfect square is 2 x 19 = 38.
76800 = 2⁸ x 3 x 5². To make this a perfect square, we need to multiply it by 2² x 5, since those are the missing factors. So the smallest number by which 76800 must be multiplied to make it a perfect square is 2² x 5 = 20.
845 = 5 x 13 x 13. To make this a perfect square, we need to multiply it by 5, since that's the missing factor. So the smallest number by which 845 must be multiplied to make it a perfect square is 5.
Therefore, the smallest numbers by which each given number must be multiplied to make it a perfect square are: 13 for 156, 2 for 432, 35 for 700, 38 for 3698, and 5 for 845. For 1331 and 882, the given numbers are already perfect squares. For 76800, the smallest number is 20.
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Answer:
We can find the smallest number by which each given number must be multiplied to make it a perfect square by first factorizing each number into its prime factors, and then identifying the missing factors that would make it a perfect square.
Let's take each number one by one:
156 = 2² x 3 x 13. To make this a perfect square, we need to multiply it by 13, since that's the missing factor. So the smallest number by which 156 must be multiplied to make it a perfect square is 13.
1331 = 11³. This is already a perfect cube, and therefore a perfect square as well. So the smallest number by which 1331 must be multiplied to make it a perfect square is 1.
432 = 2⁴ x 3³. To make this a perfect square, we need to multiply it by 2, since that's the missing factor. So the smallest number by which 432 must be multiplied to make it a perfect square is 2.
700 = 2² x 5² x 7. To make this a perfect square, we need to multiply it by 5 x 7, since those are the missing factors. So the smallest number by which 700 must be multiplied to make it a perfect square is 5 x 7 = 35.
882 = 2 x 3² x 7². This is already a perfect square, since it has an even power of each prime factor. So the smallest number by which 882 must be multiplied to make it a perfect square is 1.
3698 = 2 x 7 x 13 x 19. To make this a perfect square, we need to multiply it by 2 x 19, since those are the missing factors. So the smallest number by which 3698 must be multiplied to make it a perfect square is 2 x 19 = 38.
76800 = 2⁸ x 3 x 5². To make this a perfect square, we need to multiply it by 2² x 5, since those are the missing factors. So the smallest number by which 76800 must be multiplied to make it a perfect square is 2² x 5 = 20.
845 = 5 x 13 x 13. To make this a perfect square, we need to multiply it by 5, since that's the missing factor. So the smallest number by which 845 must be multiplied to make it a perfect square is 5.
Therefore, the smallest numbers by which each given number must be multiplied to make it a perfect square are: 13 for 156, 2 for 432, 35 for 700, 38 for 3698, and 5 for 845. For 1331 and 882, the given numbers are already perfect squares. For 76800, the smallest number is 20.
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