Question

Find the smallest number by which 2808 must be multiplied so that the product is a perfect cube.

Answer 1

Hint: The given problem is related to cubes and cube roots. Let the number be multiplied be k. So, $2808\times k$ must be a perfect cube. To determine the cube root of a number, express the number in terms of the product of its prime factors. This is the method of prime factorization.

Complete step by step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an odd number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now, coming to the question, we let the number to multiplied with 2808 to make it a perfect cube be k. Now we know that $2808\times k$ is a perfect cube. We will use the method of prime factorization. So, first we will express 2808k as the product of prime numbers.
Now, 2808k is an even number. So, we can write 2808k as $2808k=2\times 1404k$ . Now, 1404k is also an even number. So, we can write 1404k as $1404k=2\times 702k$ . Similarly, we can write 702k as $702k=2\times 351k$ . Now, we know k is also a factor so 351k can be written as $351k=351\times k$. Now, 351 is an odd number. So, it is not divisible by 2. The sum of digits of 351=3+5+1=9, which is divisible by 3. So, 351 is divisible by 3. So, we can write 351 as $351=3\times 117$ . Again, the sum of digits of 117 is 1+1+7 = 9, which is divisible by 3. So, we can write 117 as $117=3\times 39$ . Again, the sum of digits of 39 is 3+9 = 12, which is divisible by 3. So, we can write $39=3\times 13$ . We know, 13 is prime. So, we cannot further factorize it. So, $2808=2\times 2\times 2\times 3\times 3\times 3\times 13\times k$ . So, for 2808k to be a perfect cube k must have two prime factors as 13 and k will be least if $k={{13}^{2}}=169.$
Therefore, the lowest value of k for which 2808k is a perfect cube if k is equal to 169.

Note: While calculating square roots and cube roots, prime factorization is the easiest method. But it takes time. Hence, other methods should also be learnt, so that they can be used while solving problems in cases where time plays an important role.