Cubes

You know that the word ‘cube’ is used in geometry. A cube is a solid figure which has all its sides .

How many cubes of side 1 cm will make a cube of side 2 cm? 812416

How many cubes of side 1 cm will make a cube of side 3 cm ? 271896

Consider the numbers 1, 8, 27, ... These are called perfect cubes or cube numbers.

Can you say why they are named so? Each of them is obtained when a number is multiplied by taking it threetwoone times.

We note that:

1 = 1 × 1 × 1 = 1312

8 = 2 × 2 × 2 = 2322

27 = 3 × 3 × 3 = 3332

We get: 53 = 5 × 5 × 5 =

Therefore, 125 is a cubesquare number.

Is 9 a cube number? NoYes

As 9 = × and there is no natural number which multiplied by taking three times gives 9.

We can see also that 2 × 2 × 2 = and 3 × 3 × 3 = .

This shows that 9 is not a perfect cube.

The following are the cubes of numbers from 1 to 10:

There are only ten perfect cubes from 1 to 1000. How many perfect cubes are there from 1 to 100? Observe the cubes of even numbers. Are they all even? What can you say about the cubes of odd numbers?

Fill the given below table to know the cubes of the numbers from 11 to 20.

Consider a few numbers having 1 as the one’s digit (or unit’s). Find the cube of each of them. What can you say about the one’s digit of the cube of a number having 1 as the one’s digit?

Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc. Based on your observations, drop the given below numbers in the correct category:

Find the one’s digit of the cube of each of the following numbers

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Some interesting patterns

1. Adding consecutive odd numbers

Observe the following pattern of sums of odd numbers.

Is it not interesting? How many consecutive odd numbers will be needed to obtain the sum as 103?

Express the following numbers as the sum of odd numbers using the above pattern?

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Consider the following pattern:

23−13 = 1 + 2 × 1 × 3

33 – 23 = 1 + 3 × 2 × 3

43 – 33 = 1 + 4 × 3 × 3

Using the above pattern, find the value of the following:

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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.

Example 1: 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?

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Smallest multiple that is a perfect cube

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.

Example 2: 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 .

Example 3: 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

Example 4: 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

Example 5: 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 =

Think, Discuss and Write

Check which of the following are perfect cubes.

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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.