Perfect Cube & Square: 5-Digit Number

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In summary, a perfect cube is a number that can be expressed as the product of three equal numbers and a perfect square is a number that can be expressed as the product of two equal numbers. In the case of a 5-digit number, it would be a number that is both a perfect cube and a perfect square. The smallest 5-digit number that is both a perfect cube and square is 125. There are 9 possible 5-digit numbers that are both perfect cubes and squares. To determine if a 5-digit number is a perfect cube and square, you can take the cube root and square root of the number. There are patterns and rules for finding a 5-digit number that is a perfect cube and square, such
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kaliprasad
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Find 5-digit number whose half is a perfect cube and one-third is a perfect square
 
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kaliprasad said:
Find 5-digit number whose half is a perfect cube and one-third is a perfect square

$$2^{10}\cdot3^3=27648$$
 
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greg1313 said:
$$2^{10}\cdot3^3=27648$$

right ans . here is my method

the number is $2n^3a^6= 3m^2a^6$ ( we can choose m, n to be as low as possible)
now $n^3$ and $m^2$ shall be of the form $2^a3^b$ say $n = 2^x3^y$ and $m = 2^p3^q$

so we get $2^{3x+1}3^{3y} = 2^{2p} 3^{2q+1}$
comparing exponents on both sides $3x + 1 = 2p$ givinng $x = 1 , p = 2$
and $3y = 2q + 1$ giving $y = q = 1$
so $n = 2 * (2 * 3)^3 = 432$
$\frac{10000}{432} = 23$ and $\frac{100000}{432} = 231$ and 64 is the only $6^{th}$ power between 23 and 231 is $2^6 = 64$
hence $ans = 64n = 432 * 64 = 27648$
 
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FAQ: Perfect Cube & Square: 5-Digit Number

1. What is a perfect cube and square in regards to a 5-digit number?

A perfect cube is a number that can be expressed as the product of three equal numbers (ex: 125 = 5 x 5 x 5). A perfect square is a number that can be expressed as the product of two equal numbers (ex: 25 = 5 x 5). In the case of a 5-digit number, it would be a number that is both a perfect cube and a perfect square.

2. What is the smallest 5-digit number that is both a perfect cube and square?

The smallest 5-digit number that is both a perfect cube and square is 125, which is equal to 5 x 5 x 5.

3. How many 5-digit numbers are perfect cubes and squares?

There are 9 possible 5-digit numbers that are both perfect cubes and squares. These numbers are 125, 216, 343, 512, 729, 1000, 1331, 1728, and 2197.

4. How can you determine if a 5-digit number is a perfect cube and square?

To determine if a 5-digit number is a perfect cube and square, you can take the cube root and square root of the number. If the cube root is a whole number and the square root is a whole number, then the number is a perfect cube and square.

5. Are there any patterns or rules for finding a 5-digit number that is a perfect cube and square?

Yes, there are patterns and rules for finding a 5-digit number that is a perfect cube and square. One rule is that the last 3 digits of the number must be either 000, 125, 216, 343, 512, 729, 000, 331, 728, or 729. Another rule is that the sum of the digits of the number must be a multiple of 9. These rules can help narrow down the possible numbers to check for perfect cubes and squares.

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