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.
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
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............
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.
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