Question

If x is an integer greater than 1, is x equal to the 12th power of an integer ?(1) x is equal to the 3rd Power of an integer(2) x is equal to the 4th Power of an integer.

Answer 1

The statement is true only when both (1) and (2) are valid. If only one of (1) and (2) is valid, them the statement is not true.

Explanation:

(1) alone is not sufficient, 27 is 3³ but is not a 12th power

(2) alone is not sufficient either, 81 is 3³ but it is not a 12th power

If both (1) and (2) are valid, then for each prime p that divides x, p should divide y and z, with y³ = x and z⁴=x.

Lets suppose that k is the highest power of p that divides y and m is the highest power that divides z, then (p^k)³ = (p^m)⁴. Therefore

p^3k = p^4m

This means that the power of p that appears on x is a multiple of both 3 and 4. Since those numbers are coprime, then that power is a multiple of 12.

This ensures that every prime dividing x has at least a power of 12 in the prime factirization, hence x is a 12th power.