aliakberza wrote:
Thanks a lot for the explanation. Just wanted to clarify something, how do we conclude that if c is divisible by 4ab then it HAS to be divisible by ab. Could it not be the case that c is divisible by the 4a or 4b?
Say if a=3 and b=5, if c =16 then it is divisible by 4ab but not ab? Would appreciate your help in this.
If a number is divisible by d, say, then the number will also always be divisible by every factor of d. So, for example, if a number is divisible by 12, it must also be divisible by 6, 4, 3 and 2. You can see why by factoring - if a number is divisible by 12, then it equals 12q, where q is a whole number. But then it also equals 6(2q), so it's a multiple of 6 too, for example, and it also equals 4(3q), so it's a multiple of 4, and so on.
So if a number is divisible by 4ab, it is divisible by every divisor of 4ab, so it's divisible by ab, and by 4, and by 4a, and by 2b, and by several other things.
Notice in your example, where a = 3 and b = 5, then ab = 15, and 4ab = 60. That number is not divisible by 16, since 60/16 is not an integer -- it does share some divisors with 16, so you can cancel that fraction down a bit, to 15/4, but you don't get an integer when you're finished canceling.
Got it. Thanks so much for taking the time. Appreciate it!