Question

Which of the following numbers is a perfect cube?
A.1525
B.1728
C.1458
D.3993

Answer 1

Hint: When a number is multiplied by itself three times, the number obtained is called the cube of the number taken initially. The cube of an integer is called a perfect cube. For example, 125 is a perfect cube of 5, while 121 is a cube of 4.946 so it is not a perfect cube. To find out if a given number is a perfect cube or not, we factorize it and observe if it is a product of a number multiplied by itself three times.

Complete step-by-step answer:
First, we have to factorize all the numbers,
 $ 1525 = 5 \times 5 \times 61 $
1525 is the product of the square of 5 and 61, it doesn’t involve a cube of any term, so is it not a perfect cube.
 $ 1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 4 \times 4 \times 4 \times 3 \times 3 \times 3 $
It can be written as $ 1728 = 12 \times 12 \times 12 $
1728 can be written as the product of cube of 4 and 3 or we can say that it is a cube of 12.
Hence, 1728 is a perfect cube.
 $ 1458 = 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2 \times 9 \times 9 \times 9 $
1458 is the product of a cube of 9 and one 2, so it is not a perfect cube.
 $ 3993 = 3 \times 11 \times 11 \times 11 $
3993 is the product of a cube of 11 and one 3, so it is not a perfect cube.
So, the correct answer is “Option B”.

Note: The numbers that are multiplied together to form a bigger number are called the factors of that bigger number. Find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.