If the least common multiple of a positive integer x ,4^3 and 6^5 is 6

If the least common multiple of a positive integer \(x\), \(4^3\) and \(6^5\) is \(6^6\), then how many values can \(x\) take?

A. \(1\)
B. \(6\)
C. \(7\)
D. \(30\)
E. \(36\)


We are given that \(6^6=2^{6}*3^{6}\) is the least common multiple (LCM) of three numbers:

\(x\);

\(4^3=2^6\);

\(6^5 = 2^{5}*3^5\);

First notice that \(x\) cannot include any prime factors other than 2 and/or 3 as the LCM is exclusively composed of these primes.

Next, since the exponent of 3 in the LCM is greater than the exponent of 3 in the other two numbers, \(x\) must have \(3^6\) as a factor. Otherwise, the factor \(3^{6}\) wouldn't appear in the LCM.

Furthermore, \(x\) can include the prime number 2 raised to any power from 0 to 6, inclusive (the LCM limits the power of 2 in \(x\) to 6, so it cannot be any higher).

Therefore, \(x\) can take on a total of 7 distinct values, namely:

\(3^6\);

\(2*3^6\);

\(2^2*3^6\);

\(2^3*3^6\);

\(2^4*3^6\);

\(2^5*3^6\);

\(2^6*3^6\).


Answer: C