Find the smallest possible number by which 8788 must be multiplied to obtain a perfect cube.
Answer
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Hint: To find the smallest number to be multiplied by 8788 to make it a perfect cube, write the prime factorization of the number 8788. Observe the terms needed to be multiplied to it to make it a perfect cube. Multiply all the numbers that are needed to be multiplied to get the smallest number needed to make 8788 a perfect cube.
Complete step-by-step answer:
We have to find the smallest number to be multiplied to 8788 to make it a perfect cube. To do so, we will write the prime factorization of the number 8788.
To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by 3 until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers \[5,7,...\]until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.
Thus, the prime factorization of 8788 is \[8788={{2}^{2}}\times {{13}^{3}}\].
We observe that \[{{13}^{3}}\] is already a perfect cube. So, to make 8788 a perfect cube, we need to make \[{{2}^{2}}\] a perfect cube. Thus, the smallest number by which we need to multiply \[{{2}^{2}}\] is 2.
Hence, to make 8788 a perfect cube, the smallest number by which we need to multiply it is 2.
Note: It is necessary to observe that we need the smallest possible number to make 8788 a perfect cube. We will get multiple answers if we don’t consider the fact that we need to find the smallest number. A perfect cube is a number that is the cube of an integer.
Complete step-by-step answer:
We have to find the smallest number to be multiplied to 8788 to make it a perfect cube. To do so, we will write the prime factorization of the number 8788.
To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by 3 until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers \[5,7,...\]until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.
Thus, the prime factorization of 8788 is \[8788={{2}^{2}}\times {{13}^{3}}\].
We observe that \[{{13}^{3}}\] is already a perfect cube. So, to make 8788 a perfect cube, we need to make \[{{2}^{2}}\] a perfect cube. Thus, the smallest number by which we need to multiply \[{{2}^{2}}\] is 2.
Hence, to make 8788 a perfect cube, the smallest number by which we need to multiply it is 2.
Note: It is necessary to observe that we need the smallest possible number to make 8788 a perfect cube. We will get multiple answers if we don’t consider the fact that we need to find the smallest number. A perfect cube is a number that is the cube of an integer.
Last updated date: 22nd Sep 2023
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