Cubes and their Prime Factors

Consider the following prime factorisation of the numbers and their cubes.

Observe that each prime factor of a number appears threetwoone times in the prime factorisation of its cube.

In the prime factorisation of any number, if each factor appears three times then, is the number a perfect cube?

Think about it. Is 216 a perfect cube?

By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3

Each factor appears 36 times.

216 = 23×33=2×33

= 63 which is a perfect cube!

Is 729 a perfect cube? 729 = × × × × ×

YesNo, 729 isisn't a perfect cube.

Now let us check for 500?

Prime factorisation of 500 is 2 × 2 × 5 × 5 × 5 (There are three 5’s in the product but only two 2’s.)

So, 500 is notis a perfect cube.

Is 243 a perfect cube ?

243 = × × × ×

In the above factorisation 3 × 3 remains after grouping the 3’s in triplets.

Therefore, 243 is not is a perfect cube.

Which of the following are perfect cubes?

Instructions

Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm, 30 cm, 15 cm respectively.

Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?

Raj said, Volume of cuboid is

15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5

= 2 × 3 × 3 × 3 × 5 × 5 × 5

Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., to make it a perfect cube.

Therefore, we need 42 such cuboids to make a cube.

Now, is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube

392 = × × × × (Enter the factors is ascending order)

Since, the prime factor 7 does not appear in a group of three, 392 is notis a perfect cube.

To make it a cube, we need one more .

So, it becomes: 392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = which is a perfect cube.

Hence, the smallest natural number by which 392 should be multiplied to make a perfect cube is .

Is 53240 a perfect cube? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube?

53240 = × × × × × × (Enter factors in increasing order)

The prime factor 5 does not appear in a group of three. So, 53240 is notis a perfect cube.

In the factorisation 5 appears only onetwo time. If we divide the number by 5, then the prime factorisation of the quotient will not contain 5. So,

53240 ÷ 5 = 2 × 2 × 2 × 11 × 11 × 11

Hence, the smallest number by which 53240 should be divided to make it a perfect cube is .

The perfect cube in that case is i.e. 532405

Is 1188 a perfect cube? If not, by which smallest natural number should 1188 be divided so that the quotient is a perfect cube?

1188 = × × × × ×

The primes and do not appear in groups of three. So, 1188 is notis a perfect cube.

In the factorisation of 1188, the prime 2 appears only times and the prime 11 appears oncetwice.

So, if we divide 1188 by 2 × 2 × 11 = , then the prime factorisation of the quotient will not contain 2 and 11.

Hence, the smallest natural number by which 1188 should be divided to make it a perfect cube is 4422.

And the resulting perfect cube is 1188 ÷ 44 = = 33

Is 68600 a perfect cube? If not, find the smallest number by which 68600 must be multiplied to get a perfect cube.

68600 = × × × × × × × (Enter factors in increasing order)

So, 68600 is notis a perfect cube. To make it a perfect cube we multiply it by .

68600 × 5 = 2 × 2 × 2 × 5 × 5 × 5 × 7 × 7 × 7 =

Check which of the following are perfect cubes.

Instructions

What pattern do you observe in these perfect cubes ?

This pattern shows that perfect cubes often have cube roots that are whole numbers, and in cases where the number is a multiple of 10, its cube root also tends to be a neat multiple of 10.

The number of zeros in the perfect cubes is a multiple of 36.