1. Consecutive integers
For instance, you have a set of numbers: 4,5,6. The sum is 15, it is divisible by 3 (we have 3 numbers). Or consider this set: 5,6,7,8,9. The sum is 35, it's divisible by 5.
Now consider these sets: 4,5,6,7. The sum is 22, it's not divisible by 4. Or 5,6,7,8. S=26, again, not divisible by 4.
2. Prime-factorization.
In the first example the only common factor of 4 and 18 is 2^1. ( it's a factor of BOTH 4 and 18)
In the second example common factor of 4 and 24 is 2^2 (BOTH 4 and 24 are divisible by 4, we need to find maximum factor which divides evenly into both numbers)
Venn diagrams can be helpful there. For instance, we have 20 and 35. Draw 2 cirles, in one we have 2,2,5 (these are factors of 20), in other we have 5,7 (factors of 35). The only common factor is 5 - that's where our circles overlap. To find LCF we simply get ALL factors of BOTH numbers - 2,2,5,7. That's 140.
3. n! means factorial - the product of all numbers lower than n and n. For example 4!=1*2*3*4=24. 24^2 =576. Now, 4^4 (n=4 in our case) equals 256, that is less than 576. The bigger n we get, the bigger gap we have.
4. For example, we have 2 and 3. 2 is clearly less than 3, however, when we divide by 1 we change the sign: 1/2>1/3 (since we divide by LESSer amount). Always look at the signs though (whether numbers are positive or negative)
5. "-a<-b<0, -c<-d<0
Then
-a-c<-b-d<0" Hm, didn't get this part but since you are asking another issue, consider this example:
-2-5<-1-3, now, we are carrying over -5 and -3: -2-(-3)<-1-(-5) => 1<4. When we are doing this we must change the sign as we did in this example. Also notice that subtracting negative number equals to adding positive number.
6. Cancelling common terms. Actually you can meet similar equations in the real exam, that's a 500-600 level question I suppose. I've seen similar problems in
OG. Firstly, we cannot divide by x because by doing that we can incorrectly cross out one of the solutions (0 in our example). After moving x to the left side we can notice that x is a common factor of x(x-2) and x, i.e. both of these are divisible by x. So we can factor x out: x(x-2-1). When we divide x(x-2) by x we get (x-2) and when we divide x by x we get 1, minus remains. That's it I believe.
If I was wrong somewhere, correct me.