Find a positive integer m such that
1
2m is a perfect square and
1
3m is a perfect cube. Can you
find a positive integer n for which
1
2n is a perfect square,
1
3n is a perfect cube and
1
5n is a
perfect fifth power?
Please try to get into PROMYS without cheating
To find a positive integer m such that 1/2m is a perfect square and 1/3m is a perfect cube, we can start by setting up the equations:
1/2m = x^2 (1)
1/3m = y^3 (2)
Where x and y are positive integers representing the perfect square and perfect cube respectively.
From equation (1), we can rewrite it as:
1/m = 2x^2 (3)
Since m is a positive integer, 1/m must also be a positive integer. Therefore, we can conclude that m = 1/2x^2.
Substituting this value of m into equation (2), we get:
1/3(1/2x^2) = y^3
Simplifying the equation, we have:
1/(3x^2) = y^3
To find possible values of x and y, we can take the cube root of both sides:
(1/(3x^2))^(1/3) = y
Simplifying further:
1/(3^(1/3) * x^(2/3)) = y
Since y should be a positive integer, we can set the numerator to 1 and solve for x:
3^(1/3) * x^(2/3) = 1
Taking the cube of both sides to eliminate the cube root:
x^2 = 3
x = √3
Therefore, the value of m in this case is 1/2m = 1/2(√3)^2 = 1/2(3) = 3/2.
To find a positive integer n such that 1/2n is a perfect square, 1/3n is a perfect cube, and 1/5n is a perfect fifth power, we can follow a similar approach.
Setting up the equations:
1/2n = p^2 (4)
1/3n = q^3 (5)
1/5n = r^5 (6)
Where p, q, and r are positive integers representing the perfect square, perfect cube, and perfect fifth power respectively.
From equation (4), we can rewrite it as:
1/n = 2p^2 (7)
Since n is a positive integer, 1/n must also be a positive integer. Therefore, we can conclude that n = 1/2p^2.
Substituting this value of n into equations (5) and (6), we get:
1/3(1/2p^2) = q^3 (8)
1/5(1/2p^2) = r^5 (9)
Simplifying equations (8) and (9) in a similar manner as before, we will eventually need to solve the equation:
2^(2/3) * p^(4/3) = 3^(1/3) * q (10)
To ensure q is a positive integer, we can set the numerator and denominator to their respective perfect powers:
2^(2/3) * (3^(1/3))^4 = 3^(1/3) * q
(2^(2/3) * 3^(4/3))^(3/4) = q
Simplifying:
(2^2 * 3^4)^(3/4) = q
(4 * 81)^(3/4) = q
324^(3/4) = q
q = 18
Therefore, the value of n in this case is 1/2n = 1/2(18)^2 = 1/2(324) = 162.
Hence, the positive integer m is 3/2 and the positive integer n is 162.