Best Answer

648.

648/3=216 (216)^(1/3)=6

648/2=324 (324)^(1/2)=18

^^^^^ Actually, that answers a different question: "What is a number that when divided by three is a perfect cube and when divided by two is a perfect square?"

To solve the original question:

You want a number that when divided by three is a perfect square. That means it can be written as 3n2 for some integer n. You also want a number that when divided by two is a perfect cube. That means it can be written as 2m3 for some other integer m. These are two ways of writing the same number so

3n2 = 2m3

This is equivalent to

n2 = (2/3)m3

In order for the right side to be a perfect square, it must be a multiple of an even power of 2. Note that this is satisfied if m is a multiple of an odd power of 2. The right side must also be a multiple of an even power of 3. This is satisfied if m is a multiple of an odd power of 3. That is, m needs to be of the form:

m = 22k+132j+1 for j, k non-negative integers.

(As an aside, notice that three possible values of m are 6, 24, and 54.)

Going back to the original problem, the smallest number that satisfies the conditions comes from m = 6:

2 x 63 = 432. (Note that 432 / 3 = 144 = 122).

But there are infinitely many numbers that work. For example

2 x 243 = 27648. (Note that 27648 / 3 = 962).

Q: What is the smallest positive integer that is a perfect square when divided by three and it is a perfect cube when divided by two?

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324

The integer is 26

In terms of prime factors, 1008 = 24*32*7 Then since 24 and 32 are perfect squares, all that is required is to make 7q a perfect square and so q = 7.

The answer is 19.

The smallest integer is 2.There is no smallest number. Dividing 1152 by any number of the form x = 1/(2k2) where k is an integer will result in a perfect square. Since there is no limit to how large k can be, there is no limit to how small x can be.

Related questions

6.

324

5

The integer is 26

the answer is 144, it is divisible by 1, 4, 9, 16, 36, and 144.

63= 9* 7. 9 is already a perfect square, so mulitiply by 7. 7 is your answer.

its 81

The prime factorisation of 248832 is 2¹⁰ × 3⁵ Every perfect square number has a prime factorisation where each prime is to an even power. 2 has an even power 3 has an odd power, so need an extra power → multiple 248832 by 3 which gets (2⁵ × 3³)²

In terms of prime factors, 1008 = 24*32*7 Then since 24 and 32 are perfect squares, all that is required is to make 7q a perfect square and so q = 7.

17

The answer is 19.

It is a natural number. It is a positive integer. It is a positive rational number. It is a positive real number. It is a perfect square. It is a three digit integer. It is a palindromic integer. Probably many other sorts.