6.2.2. 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?

In this chunk, we discuss the dimensions of a cuboid – specifically, its length, breadth, and height. A cuboid can be visualized as a box with three dimensions. The volume of a cuboid is calculated by multiplying these three dimensions. For example, the volume is derived as Volume = Length × Breadth × Height, which in this case equals 15 cm * 30 cm * 15 cm.

Consider packing boxes for a shipping company. Each box has a specific size, like Raj's cuboid. Just as the shipping company needs to understand the volume of each box, Raj needs to know how many cuboids will fit into a perfect cube for his project.

Raj said, Volume of cuboid is 15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5 = 2 × 3 × 3 × 3 × 5 × 5 × 5.

Here, Raj calculates the volume of his cuboid, which involves breaking down the dimensions into their prime factors. This mathematical technique is valuable because it allows us to express the number in terms of its prime factors, making it easier to determine if the volume is a perfect cube.

Imagine you are organizing items into smaller boxes. Each box's contents can be referred to by factoring them down to their simplest elements. This helps you see how many boxes fit into a larger storage unit, similar to how Raj is trying to see how his cuboids fit into a perfect cube.

Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make it a perfect cube. Therefore, we need 4 such cuboids to make a cube.

To transform the volume of the cuboid into a perfect cube, we require every prime factor to appear in multiples of three. Since the factor '2' appears only once, we will need to add two more factors of '2' to form a full triplet. Therefore, we conclude that four cuboids will provide the necessary factors to create a perfect cube.

Think of a recipe that calls for groups of three ingredients. If you only have one ingredient, you can't complete the recipe until you gather more. Similarly, Raj needs to gather enough cuboids to complete his perfect cube 'recipe'.

Example 2: 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. Solution: 392 = 2 × 2 × 2 × 7 × 7.

In this chunk, we analyze whether the number 392 is a perfect cube. By factorizing it, we find that it can be represented by prime factors. To determine if a number is a perfect cube, we look for each prime factor to occur in groups of three.

Imagine a sports team where each player has a role – if there aren't enough players in each role, the team can't function. Similarly, for 392 to be a perfect cube, it needs the right number of each prime factor (players) to be organized in the correct group (three at a time).

To make its a cube, we need one more 7. In that case 392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube. Hence the smallest natural number by which 392 should be multiplied to make a perfect cube is 7.

After determining that 392 is not a perfect cube, we calculate that multiplying it by 7 introduces another factor of 7, thereby allowing every prime number to occur in groups of three. The result, 2744, is a perfect cube. Therefore, 7 is identified as the smallest natural number needed to convert 392 into a perfect cube.

Think of adding more chairs to a dining table setting. If you need all sets of three for a proper arrangement, then adding one more chair (or factor) completes your setup. Here, multiplying by 7 completes the necessary setup for a perfect cube.

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