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# Number Theory Help

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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.

Dec 13, 2019

#3
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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

Dec 13, 2019

#1
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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............

Dec 13, 2019
#2
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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.

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
#4
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Thank you.

Dec 14, 2019