Alan

Factor n^3 - n to get n(n^2 - 1) then n(n - 1)(n + 1) using the diference of two squares.  Hence the largest integer that divides n^3 - n is n + 1.

Define E(a,b,c) = a . b^2 + c. What value of a is the solution to the equation 

E(a,4,5) = E(a,6,7)?

a*4² +  5 = a*6² + 7 

16a + 5 = 36a + 7

5 - 7 = 36a - 16a

20a = -2

a = -1/10  or a =- 0.1

As follows:

I'll leave you to crunch the numbers.

You can only cancel x+3 from borh sides (i.e. divide both sides by x+3) as long as x+3 is not zero.  Here, x+3 turns out to be zero.

Fortuitously, you still get the right answer in this case!  However, performing mathematically illegal operations is not valid, whatever the apparent result. 

I get the following:

Use Pythagoras' theorem to find the length of the hypotenuse, then use the definitions of sin, cos and tan

i.e sin = opposite/hypotenuse

    cos = adjacent /hypotenuse

    tan = opposite/adjacent

ABC and DBE are similar triangles so  DB/AB = EB/CB

(10-x)/(10-x+x) = 3/(3+2)   or  (10 - x)/10 = 3/5 

Hence  10 - x = 30/5

            10 - x = 6

            x = 4

Here's another way of doing this for those who don't have a handy facility for factorising large numbers:

Let the number be ab

10a + b = 3(a + b)

10b + a = 3(10a + b) - 9

Use these two equations to find a and b.

Boys age = B

Sister's age = S

B = 2*S

B - 5 = 3*(S - 5)

Use these two equations to find B and S.