What is the value of : 16*67*45/45(65+67*65)

Here's another way to solve this
problem:

There is a common factor of 45 in the numerator
and denominator; those divide out to equal 1.  Then we have
(16*67)/(65+67*65).

Although there is a 67 in both the
numerator and denominator, this is NOT a common factor, because there is something else
added to the product in the denominator.  So we have to leave it as is and do some
multiplication.  16*67 = 1072, and 67*65 = 4355, then we add the 65 to get
4420.

Now, our fraction is 1072/4420.  If you know the
rules for divisibility by 4 (if the last two digits of a number are a number that is a
multiple of 4, then the entire number is a multiple of 4.  [see below the
answer for an explanation of why that works.]

Well, 72 = 4*18, so 1072 has a
factor of 4. (We also know that's true because one of the factors was 16, or 4*4.)  20 =
4*5, so 4420 also is divisible by 4.  1072 = 4*268 and 4420=4*1105, so the
simplified fraction = 268/1105.  This fraction doesn't simplify further
because 268=2*2*67 and
1105=5*221.

Now, why does the
divisibility rule for 4's work? Obviously, if a two-digit number is
divisible by 4, that takes care of the divisibility of that part of the number.  100 is
divisible by 4 (e.g., 4 quarters = $1.00); therefore, any multiple of 100--including
1,000 (10^3), 10^4, etc.--will be divisible by 4.