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Find the smallest multiple of 3,4,6 which is a perfect cube.(solution sholud be elaborated)

Find the smallest multiple of 3,4,6 which is a perfect cube.
Here, we need to find the smallest of 3,4 and 6 such that the number is a perfect cube.
For this, first find the L.C.M of 3,4 and 6. So,
L.C.M of 3,4 and 6 = 2*2*3
  = 12
Now, for the number to be a perfect cube, the factors of that number should appear as a multiple of 3 i.e. 3 times, 6 times, 9 times and so on.
So, prime factorizations of 12 = 2*2*3
Next, we need to multiply the factors of 12 with one 2 and two 3 in order to make it a perfect cube. So,
2*2*2*3*3*3 = 216
Now it can be checked that 216 is a multiple of 3, 4 and 6.
216/3=72
216/4=54
216/6=36

Therefore, the smallest multiple of 3, 4 and 6 such that it is a perfect cube is 216

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View Full Answer

First find the L.C.M. of 3,4, and 6 by prime factorisation method. The L.C.M. is 12.

Now by using prime factorisation method, find the prime factors of 12. They are 2,2, and 3.

2 appears twice and 3 appears once. But to make the number a perfect cube, the factors should appear 3 times or as many times as the multiples of 3 i.e. it can appear 3,6,9,12 times, etc.

So the number should be multiplied with another 2 and two more 3s to become a perfect cube i.e. the final number is

2*2*3*2*3*3 = 12*18 = 216 which is the cube of 6.

Thus 216 is the smallest multiple of 3,4, and 6 which is a perfect cube.

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