Question

Find the smallest numbers by which each of the following numbers must be divided to obtain a perfect cube:
A) $ 1536 $
B) $ 10985 $
C) $ 28672 $
D) $ 13718 $

Answer 1

Hint: In this question, we need to find the smallest numbers by which each of the given numbers must be divided to obtain a perfect cube. To find the smallest number, we will use the prime factorization method. And, group the factors into triplets and find the factor that is left. The factor that is left is the smallest number that must be divided.

Complete step-by-step answer:
A) The given value here is $ 1536 $ .
By prime factorization,
 $ 1536 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 $
Now, to obtain a perfect cube, group the factors into triplets i.e., to the power of $ 3 $ .
 $ 1536 = {2^3} \times {2^3} \times {2^3} \times 3 $
Now, by grouping we have one factor $ 3 $ , is left.
Therefore, $ 1536 $ is not a perfect cube.
So, now to make $ 1536 $ as a perfect cube, we have to divide by $ 3 $ .
Hence, the smallest number by which $ 1536 $ must be divided to obtain a perfect cube is $ 3 $ .

B) The given value is $ 10985 $ .
Similarly, by prime factorization,
 $ 10985 = 5 \times 13 \times 13 \times 13 $
Now, to obtain a perfect cube, group the factors into triplets i.e., to the power of $ 3 $ .
 $ 10985 = 5 \times {13^3} $
Now, by grouping we have one factor $ 5 $ , is left.
Therefore, $ 10985 $ is not a perfect cube.
So, now to make $ 10985 $ as a perfect cube, we have to divide by $ 5 $ .
Hence, the smallest number by which $ 10985 $ must be divided to obtain a perfect cube is $ 5 $ .

C) The given value is $ 28672 $ .
Similarly, by prime factorization,
 $ 28672 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 $
Now, to obtain a perfect cube, group the factors into triplets i.e., to the power of $ 3 $ .
 $ 28672 = {2^3} \times {2^3} \times {2^3} \times {2^3} \times 7 $
Now, by grouping we have one factor $ 7 $ , is left.
Therefore, $ 28672 $ is not a perfect cube.
So, now to make $ 28672 $ as a perfect cube, we have to divide by $ 7 $ .
Hence, the smallest number by which $ 28672 $ must be divided to obtain a perfect cube is $ 7 $ .

D) The given value is $ 13718 $ .
Similarly, by prime factorization,
 $ 13718 = 2 \times 19 \times 19 \times 19 $
Now, to obtain a perfect cube, group the factors into triplets i.e., to the power of $ 3 $ .
 $ 13718 = 2 \times {19^3} $
Now, by grouping we have one factor $ 2 $ , is left.
Therefore, $ 13718 $ is not a perfect cube.
So, now to make $ 13718 $ as a perfect cube, we have to divide by $ 2 $ .
Hence, the smallest number by which $ 13718 $ must be divided to obtain a perfect cube is $ 2 $

Note: In this question, it is important to note that a number is a perfect cube only when each factor in the prime factorization is grouped in triples. Be careful during prime factorization. Prime factorization is also known as prime decomposition. And, it is a method of breaking a number down into the set of prime numbers which multiply together to result in the original number.