H.C.F. of two numbers is always a factor of their L.C.M. ( True/False) ?

True.

The statement is true. H.C.F. stands for "Highest Common Factor," and L.C.M. stands for "Least Common Multiple."

Let's understand why the statement is true:

Let two numbers be a and b. Their H.C.F. is the greatest number that divides both a and b without leaving any remainder. Their L.C.M. is the smallest number that is a multiple of both a and b.

Let H be the H.C.F. of a and b, and let L be the L.C.M. of a and b.

Since H is the highest common factor of a and b, it can be expressed as a product of some prime factors raised to certain powers, i.e., H = p^m * q^n * ...

Since L is the least common multiple of a and b, it is also a product of prime factors raised to certain powers, i.e., L = p^x * q^y * ...

Since H.C.F. and L.C.M. are both divisible by the same prime factors (though possibly raised to different powers), it follows that H.C.F. is a factor of L.C.M. For example, if H.C.F. has the factors p^m, q^n, etc., then L.C.M. must have at least those factors (and possibly more due to additional factors raised to different powers), making H.C.F. a factor of L.C.M.

Therefore, the statement "H.C.F. of two numbers is always a factor of their L.C.M." is true.