Question

15. Using prime factorisation, check which of the following are perfect cubes?

a) 6125 b) 10584 c) 16000
16. Is 98304 a perfect cube? If not, find the smallest number by which it should be divided
to get a perfect cube.

Answer 1

Let's use prime factorization to check whether the given numbers are perfect cubes and determine if 98304 is a perfect cube.

a) 6125:
The prime factorization of 6125 is 5 * 5 * 5 * 7 * 7. We can see that it is not a perfect cube because the powers of the prime factors are not all multiples of 3. Therefore, 6125 is not a perfect cube.

b) 10584:
The prime factorization of 10584 is 2 * 2 * 2 * 3 * 13 * 17. Similar to the previous case, the powers of the prime factors are not all multiples of 3. Hence, 10584 is not a perfect cube.

c) 16000:
The prime factorization of 16000 is 2 * 2 * 2 * 2 * 2 * 5 * 5 * 5. Here, we can see that the powers of all the prime factors are multiples of 3. Therefore, 16000 is a perfect cube.

Now, let's move on to 98304:

The prime factorization of 98304 is 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3. We can observe that the power of the prime factor 3 is not a multiple of 3. Therefore, 98304 is not a perfect cube.

To find the smallest number by which 98304 should be divided to get a perfect cube, we need to divide it by the prime factors with the minimum powers until we obtain a perfect cube.

98304 divided by 2 * 2 * 2 * 2 * 2 * 2 equals 768.
768 divided by 2 * 2 * 2 equals 96.
96 divided by 2 * 2 equals 24.
24 divided by 2 * 2 equals 6.

Finally, 6 divided by 2 equals 3.

Hence, the smallest number by which 98304 should be divided to get a perfect cube is 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 = 2^10 * 3^3 = 1728.

Answer 2

Let's first find the prime factorization of each number to determine if they are perfect cubes:

a) 6125:
Prime factorization of 6125 = 5 * 5 * 5 * 7 * 11
It's not a perfect cube because the exponents of the prime factors are not all multiples of 3.

b) 10584:
Prime factorization of 10584 = 2 * 2 * 2 * 3 * 13 * 17
It's not a perfect cube because the exponents of the prime factors are not all multiples of 3.

c) 16000:
Prime factorization of 16000 = 2 * 2 * 2 * 2 * 2 * 5 * 5 * 5 * 5
It is a perfect cube because each prime factor is raised to a power that's a multiple of 3: 2^6 * 5^4.

So, the perfect cube among these numbers is 16000.

Now, for question 16:

To determine if 98304 is a perfect cube, let's find its prime factorization:
Prime factorization of 98304 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3
It is a perfect cube because each prime factor is raised to a power that's a multiple of 3.

So, 98304 is already a perfect cube.

There's no need to divide it by anything to make it a perfect cube because it already is one.