anceer wrote:

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

A 1

B 6

C 7

D 30

E 36

\(4^3=2^6\), \(6^5= 2^5.3^5\)

now if the power of 3 in x is less than 6, then l.c.m will have \(3^5\)in it.

if the power of 3 in x is more than 6, then l.c.m will have a term more than \(3^6\).

thus x must have \(3^6\) in it, and x cannot have any prime number other than 2 and 3. why ?? because in that case l.c.m won't be \(6^6\).

Also, maximum power of 2 that x can contain is 6. thus power of 2 in x can vary from 2^0 to 2^6. this gives us 7 possible cases. which are \(2^0.3^6,2^1.3^6,2^2.3^6,2^3.3^6,2^4,3^6,2^5.3^6,2^6.3^6\)

hence answer is C