Of all the 900,000 6-digit integers, how many of them are both perfect squares and perfect cubes for the same number? Thanks for help.

Guest Dec 13, 2019

#3**+1 **

If I understand the question:

A number is perfect square and a perfect cube if it follows this rule: If its prime factor(s) is a multiple of 6. So, you would try the first few small primes raised to power of: 6, 12, 18.......etc:

2^6 = 64 too small. 2^12 =4096 still too small. 2^18 =262144 - just right! so that one number.

3^6 =729 too small. 3^12 =531441 - just right. So that is your 2nd number.

5^6 =15625 - too small. 5^12 =244,140,625 way too large.

7^6 =117649 - just right. So, that is your 3rd number.

11^6 =1,771,561 - too large.

And that is it. You have ONLY 3 such numbers:

Number Sq.root Cubic root

117649 343 49

262144 512 64

531441 729 81

Guest Dec 13, 2019

#1**0 **

How can they be perfect squres and pefect cubes of the __SAME__ number?????

Do you mean perfect squares and perfect cubes of DIFFERENT numbers?...or something else?

BTW...I cannot answer this Q............

ElectricPavlov Dec 13, 2019

#2**0 **

Sorry EP: I mean the same 6-digit number is a perfect square AND a perfect cube of 2 different numbers such as 4096 which is a perfect square of 64 and a perfect cube of 16.

Guest Dec 13, 2019

#3**+1 **

Best Answer

If I understand the question:

A number is perfect square and a perfect cube if it follows this rule: If its prime factor(s) is a multiple of 6. So, you would try the first few small primes raised to power of: 6, 12, 18.......etc:

2^6 = 64 too small. 2^12 =4096 still too small. 2^18 =262144 - just right! so that one number.

3^6 =729 too small. 3^12 =531441 - just right. So that is your 2nd number.

5^6 =15625 - too small. 5^12 =244,140,625 way too large.

7^6 =117649 - just right. So, that is your 3rd number.

11^6 =1,771,561 - too large.

And that is it. You have ONLY 3 such numbers:

Number Sq.root Cubic root

117649 343 49

262144 512 64

531441 729 81

Guest Dec 13, 2019