The smallest number which should be multiplied to 432 to get a perfect cube is ____.

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Detailed Solution

Concept- 4 should be multiplied to 432 to get a perfect cube.
A number called the "perfect cube" has all three of its prime factors.
Given number is 432
(2) Therefore, prime factors of 432 are 2×2×2×3×3×3×22×2×2×3×3×3×2=

2^{4} \times 3^{3}

(3) Writing the prime factor in a trio=

2^{4} \times 3^{3}

(4) We can see from the step above that 2 is the only number that is not part of the trio.
So, 432 is not a perfect cube number, we can claim.
(5) Now to make 432 as a perfect cube number, we should multiply it with such a number that it completes the three groups of elements of 2. (i.e., by 4)
(6) Therefore, on multiplying 432 by 4, we get prime factors as

2^{6} \times 3^{3}

2^{3} \times 2^{3} \times 3^{3}

12^{3}

=12×12×12=12×12×12
=1728
We can observe that a group of three elements has all of the aforementioned prime factors. The number is a perfect cube as a result.
(7) In order to transform 432 into a perfect cube, the least amount by which it must be multiplied is 4. A perfect cube is 1728.
Hence, the answer is 4.

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